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Trigonometry","subjectCode":"a_level_maths__aqa","surface":"questions"}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"LearningResource\",\"name\":\"Question 308 – 1.8 E: Trigonometry - A Level Maths Questions\",\"description\":\"Practise 1.8 E: Trigonometry A Level Maths exam-style questions with worked solutions on MathsGenie.\",\"learningResourceType\":\"Practice questions\",\"educationalLevel\":\"A Level\",\"about\":{\"@type\":\"Thing\",\"name\":\"1.8 E: Trigonometry\"},\"url\":\"https://mathsgenie.co.uk/a-level/maths/aqa/e-trigonometry/questions/q308\",\"inLanguage\":\"en-GB\",\"provider\":{\"@id\":\"https://mathsgenie.co.uk/#organization\"}}"}}],["$","div",null,{"className":"min-h-0 w-full flex-1 overflow-hidden bg-background","children":["$","$L39",null,{"backHref":"/a-level/maths/aqa/questions","basePath":"/a-level/maths/aqa/e-trigonometry/questions","formulaUrl":"https://filestore.aqa.org.uk/resources/mathematics/AQA-AS-A-MATHS-FORMULAE.PDF","initialIndex":307,"initialQuestionWindow":[{"id":"f6de1d5e-522e-4ccf-a658-ef0c2f5a78dd","revisionId":"4b72e7e6-4a44-4c74-9ce0-5116444fccca","questionNumber":null,"questionType":"LA","specification":"An offshore wave-energy converter consists of a floating buoy and a submerged dampener plate that both move vertically due to ocean swells. Over a specific operating cycle, the height $h_B$ metres of the buoy above the seabed at time $t$ seconds is given by\n\\[h_B = 50 - 7\\sin t\\]\nThe height $h_D$ metres of the dampener plate above the seabed at time $t$ seconds is given by\n\\[h_D = 15 - 4\\cos t\\]","difficultyBand":"medium","tier":null,"totalMarks":8,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"ae50bae1-e39a-4af8-872b-20077213ce87","label":"a","content":"Show that the initial vertical distance between the buoy and the dampener plate is $39$ metres.","markscheme":"- Substitutes $t = 0$ into both equations to find $h_B = 50 - 7\\sin(0) = 50$ and $h_D = 15 - 4\\cos(0) = 11$ M1\n- Calculates the difference $50 - 11 = 39$ m A1","marks":2,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"individual_evaluation","label":"Individual Evaluation","steps":[{"type":"text","order":0,"title":"Understand \"initial\" height","content":"To find the initial position of any object in a time-dependent model, we look at the start of the time period, which corresponds to $$t = 0$$.\n\nWe will evaluate the heights of the buoy ($h_B$) and the dampener plate ($h_D$) individually at this time.","highlights":[{"text":"initial vertical distance","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$h_B = 50 - 7\\sin(\\htmlData{anchor=t-val-b}{\\color{@sky}{0}})$"},{"latex":"$$h_B = 50 - 7(\\htmlData{anchor=sin-val}{\\color{@sky}{0}})$"},{"latex":"$$h_B = \\htmlData{anchor=hb-final}{\\color{@green}{50}} \\text{ m}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"t-val-b"},{"color":"green","style":"box","anchor":"hb-final"}],"connections":[{"to":"sin-val","from":"t-val-b","color":"sky","label":"sin(0) = 0","style":"solid"}]},"order":1,"title":"Evaluate the buoy height","description":"Substitute $t = 0$ into the equation for the buoy height, $h_B = 50 - 7\\sin t$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$h_D = 15 - 4\\cos(\\htmlData{anchor=t-val-d}{\\color{@sky}{0}})$"},{"latex":"$$h_D = 15 - 4(\\htmlData{anchor=cos-val}{\\color{@sky}{1}})$"},{"latex":"$$h_D = 15 - 4$"},{"latex":"$$h_D = \\htmlData{anchor=hd-final}{\\color{@amber}{11}} \\text{ m}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"t-val-d"},{"color":"amber","style":"box","anchor":"hd-final"}],"connections":[{"to":"cos-val","from":"t-val-d","color":"sky","label":"cos(0) = 1","style":"solid"}]},"order":2,"title":"Evaluate dampener plate height","description":"Now, substitute $t = 0$ into the equation for the dampener plate height, $h_D = 15 - 4\\cos t$."},{"type":"text","order":3,"title":"Calculate the distance","content":"The vertical distance between the buoy and the dampener plate is the difference between their respective heights:\n\n$$\\text{Vertical Distance} = h_B - h_D$$\n\nSubstituting our calculated values:\n\n$$\\text{Vertical Distance} = 50 - 11 = 39 \\text{ m}$$\n\nThis confirms the initial vertical distance is indeed **$39$ metres**.","highlights":[{"text":"39","location":"specification"}]}]},{"id":"distance_function","label":"Distance Function Formulation","steps":[{"type":"text","order":0,"title":"Define the distance function","content":"To find the vertical distance between the buoy and the dampener plate at any time $t$, we can define a single distance function, $d(t)$, representing the difference between their heights:\n\n$$d(t) = h_B(t) - h_D(t)$$\n\nSubstituting the given expressions for $h_B$ and $h_D$:\n\n$$d(t) = (50 - 7\\sin t) - (15 - 4\\cos t)$$\n\nThis single function gives the vertical distance between the two components at any time $t$ in seconds.","highlights":[{"text":"h_B = 50 - 7\\sin t","location":"specification"},{"text":"h_D = 15 - 4\\cos t","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$d(t) = 50 - 7\\sin t - 15 + 4\\cos t$"},{"latex":"$$d(t) = \\htmlData{anchor=const}{\\color{@sky}{35}} - 7\\sin t + 4\\cos t$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"const"}],"connections":[]},"order":1,"title":"Simplify the function","description":"We simplify the expression by combining the constant numerical terms: \n\n$$50 - 15 = 35$$\n\nThis gives us the simplified distance function:\n\n$$d(t) = 35 - 7\\sin t + 4\\cos t$$"},{"type":"text","order":2,"title":"Find the initial distance","content":"The \"initial\" distance refers to the distance at the very start of the operating cycle, which is when time $t = 0$.\n\nSubstitute $t = 0$ into our simplified distance function $d(t)$:\n\n$$d(0) = 35 - 7\\sin(0) + 4\\cos(0)$$\n\nRecall the exact values for the trigonometric functions at $0$:\n- $\\sin(0) = 0$\n- $\\cos(0) = 1$\n\nNow substitute these values in:\n\n$$\\begin{aligned} d(0) &= 35 - 7(0) + 4(1) \\\\ d(0) &= 35 + 4 \\\\ d(0) &= 39 \\text{ metres} \\end{aligned}$$\n\nThis matches the expected answer and earns both marks under the markscheme."}]}],"generatedAt":"2026-06-11T03:04:44.695Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"2dd37b01-45d7-4392-b31d-eb55a13b7302","label":"b","content":"Show that the distance $d$ metres between the buoy and the dampener plate at time $t$ is given by\n\\[d = 35 + R\\cos(t + \\alpha)\\]\nwhere $R$ and $\\alpha$ are positive constants to be found. Give $\\alpha$ in radians to two decimal places.","markscheme":"- Subtracts height expressions to find $d = (50 - 7\\sin t) - (15 - 4\\cos t) = 35 + 4\\cos t - 7\\sin t$ B1\n- Attempts to find $R$ using $R = \\sqrt{4^2 + 7^2}$ M1\n- Attempts to find $\\alpha$ using $\\tan \\alpha = \\frac{7}{4}$ M1\n- Obtains $R = \\sqrt{65}$ (or approximately $8.06$) and $\\alpha = 1.05$ (allow $1.05$ only) A1","marks":4,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"compound_angle_expansion","label":"Compound Angle Expansion","steps":[{"type":"text","order":0,"title":"Subtract the height expressions","content":"To find the distance $d$ between the buoy and the dampener plate at time $t$, we subtract the expression for the height of the dampener plate $h_D$ from the height of the buoy $h_B$:\n\n$$d = h_B - h_D$$\n\nSubstituting our given expressions:\n\n$$d = (50 - 7\\sin t) - (15 - 4\\cos t)$$\n$$d = 50 - 15 + 4\\cos t - 7\\sin t$$\n$$d = 35 + 4\\cos t - 7\\sin t$$\n\nThis earns the first mark **(B1)**.","highlights":[{"text":"h_B = 50 - 7\\sin t","location":"specification"},{"text":"h_D = 15 - 4\\cos t","location":"specification"}]},{"type":"text","order":1,"title":"Use compound angle expansion","content":"We need to express $4\\cos t - 7\\sin t$ in the form $R\\cos(t + \\alpha)$.\n\nUsing the compound angle identity for cosine:\n\n$$\\cos(A + B) = \\cos A \\cos B - \\sin A \\sin B$$\n\nExpanding $R\\cos(t + \\alpha)$ gives us:\n\n$$R\\cos(t + \\alpha) = R\\cos t \\cos \\alpha - R\\sin t \\sin \\alpha$$\n\nWe now equate this expansion to our expression $4\\cos t - 7\\sin t$:\n\n$$R\\cos t \\cos \\alpha - R\\sin t \\sin \\alpha = 4\\cos t - 7\\sin t$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(\\htmlData{anchor=lh1}{\\color{@sky}{R \\cos \\alpha}})\\cos t - (\\htmlData{anchor=lh2}{\\color{@amber}{R \\sin \\alpha}})\\sin t = \\htmlData{anchor=rh1}{\\color{@sky}{4}}\\cos t - \\htmlData{anchor=rh2}{\\color{@amber}{7}}\\sin t$"},{"latex":"$$$R \\cos \\alpha = 4 \\quad \\text{and} \\quad R \\sin \\alpha = 7$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"lh1"},{"color":"sky","style":"text-color","anchor":"rh1"},{"color":"amber","style":"text-color","anchor":"lh2"},{"color":"amber","style":"text-color","anchor":"rh2"}],"connections":[{"to":"rh1","from":"lh1","color":"sky","label":"equate cos","style":"solid"},{"to":"rh2","from":"lh2","color":"amber","label":"equate sin","style":"solid"}]},"order":2,"title":"Equate the coefficients","description":"By comparing the coefficients of $\\cos t$ and $\\sin t$ on both sides of the equation, we get two simultaneous equations."},{"type":"text","order":3,"title":"Solve for R","content":"To find $R$, we square and add our two equations:\n\n$$(R \\sin \\alpha)^2 + (R \\cos \\alpha)^2 = 7^2 + 4^2$$\n$$R^2 (\\sin^2 \\alpha + \\cos^2 \\alpha) = 49 + 16$$\n\nSince $\\sin^2 \\alpha + \\cos^2 \\alpha = 1$, we get:\n\n$$R^2 = 65$$\n$$R = \\sqrt{65} \\approx 8.06$$\n\nThis step earns the second mark **(M1)**."},{"type":"text","order":4,"title":"Solve for alpha","content":"To find $\\alpha$, we divide the sine equation by the cosine equation:\n\n$$\\frac{R \\sin \\alpha}{R \\cos \\alpha} = \\frac{7}{4}$$\n$$\\tan \\alpha = \\frac{7}{4}$$\n\nThis earns the third mark **(M1)**.\n\nNow we calculate $\\alpha$ in radians:\n\n$$\\alpha = \\arctan\\left(\\frac{7}{4}\\right) \\approx 1.05166... \\approx 1.05 \\text{ rad}$$\n\nRounding to two decimal places gives $\\alpha = 1.05$ **(A1)**.","calculator":[{"gdc":{"expression":"arctan(7/4)","instructions":"1. Check that the angle settings are set to Radians.\n2. Access the inverse tangent function: press [SHIFT] [TAN] or locate 'arctan'.\n3. Input 7/4 and execute."},"ti84":{"expression":"tan⁻¹(7/4)","instructions":"1. Ensure your calculator is in Radian mode by pressing [MODE], highlighting 'RADIAN', and pressing [ENTER].\n2. Press [2nd] [TAN] to select $\\tan^{-1}$.\n3. Type 7 / 4 and press [ENTER]."},"desmos":{"expression":"\\arctan\\left(\\frac{7}{4}\\right)","instructions":"1. Set the grid to 'Radians' mode in the Graph Settings (wrench icon).\n2. Type 'arctan(7/4)' in the input bar to get the result."},"description":"Calculate the angle $\\alpha$ in radians using your calculator."}]},{"type":"text","order":5,"title":"State the final expression","content":"Now we substitute the values of $R$ and $\\alpha$ back into our expression for the distance $d$:\n\n$$d = 35 + \\sqrt{65}\\cos(t + 1.05)$$\n\nwhere $R = \\sqrt{65}$ (or $\\approx 8.06$) and $\\alpha = 1.05$ (to $2$ decimal places)."}]},{"id":"direct_formula","label":"Direct R-Formula Application","steps":[{"type":"text","order":0,"title":"Set up distance equation","content":"The distance $d$ between the floating buoy and the submerged dampener plate is found by subtracting the height of the plate, $h_D$, from the height of the buoy, $h_B$:\n\n$$d = h_B - h_D$$ \n\nSubstituting the given expressions:\n\n$$d = (50 - 7\\sin t) - (15 - 4\\cos t)$$\n\n$$d = 50 - 7\\sin t - 15 + 4\\cos t$$\n\n$$d = 35 + 4\\cos t - 7\\sin t$$","highlights":[{"text":"h_B = 50 - 7\\sin t","location":"specification"},{"text":"h_D = 15 - 4\\cos t","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{We compare } \\htmlData{anchor=expr}{\\color{@sky}{4\\cos t - 7\\sin t}} \\text{ to the general R-formula format:}$"},{"latex":"$$$a\\cos t - b\\sin t = R\\cos(t + \\alpha)$$"},{"latex":"$$\\text{By matching coefficients, we identify:}$"},{"latex":"$$$\\htmlData{anchor=a-val}{\\color{@purple}{a = 4}} \\quad \\text{and} \\quad \\htmlData{anchor=b-val}{\\color{@teal}{b = 7}}$$"}],"highlights":[{"color":"sky","style":"box","anchor":"expr"},{"color":"purple","style":"text-color","anchor":"a-val"},{"color":"teal","style":"text-color","anchor":"b-val"}],"connections":[{"to":"a-val","from":"expr","color":"purple","label":"cosine coefficient","style":"solid"},{"to":"b-val","from":"expr","color":"teal","label":"sine coefficient","style":"solid"}]},"order":1,"title":"Apply the R-formula","description":"We match the coefficients of our equation to the standard R-formula."},{"type":"text","order":2,"title":"Calculate amplitude R","content":"The constant $R$ represents the amplitude of the combined trigonometric function and is given directly by:\n\n$$R = \\sqrt{a^2 + b^2}$$\n\nSubstituting $a = 4$ and $b = 7$:\n\n$$R = \\sqrt{4^2 + 7^2}$$\n\n$$R = \\sqrt{16 + 49} = \\sqrt{65} \\approx 8.06$$"},{"type":"text","order":3,"title":"Calculate phase angle alpha","content":"The phase angle $\\alpha$ is determined directly using the relation:\n\n$$\\tan\\alpha = \\frac{b}{a}$$\n\nSubstituting our values:\n\n$$\\tan\\alpha = \\frac{7}{4}$$\n\n$$\\alpha = \\arctan\\left(\\frac{7}{4}\\right)$$\n\nUsing a calculator in radian mode:\n\n$$\\alpha \\approx 1.05165... \\text{ rad}$$ \n\nRounding to two decimal places, we get:\n\n$$\\alpha \\approx 1.05$$","calculator":[{"gdc":{"expression":"arctan(7/4)","instructions":"Set angle mode to Radians. Navigate to the run-matrix screen and calculate the inverse tangent of 7/4."},"ti84":{"expression":"tan⁻¹(7/4)","instructions":"Ensure your calculator is in RADIAN mode (Press [MODE], select 'RADIAN', and press [ENTER]). Then press [2nd] [TAN] ( 7 / 4 ) [ENTER]."},"desmos":{"expression":"\\arctan(7/4)","instructions":"Ensure Desmos is set to Radians. Type 'arctan(7/4)' or use the keypad to find the inverse tangent."},"description":"Calculate the inverse tangent of 1.75 to find the angle in radians."}]},{"type":"text","order":4,"title":"Write final expression","content":"Now we substitute $R$ and $\\alpha$ back into our overall expression for the distance $d$:\n\n$$d = 35 + R\\cos(t + \\alpha)$$\n\nWhere:\n- $R = \\sqrt{65}$ (or $\\approx 8.06$)\n- $\\alpha = 1.05$ (to $2$ d.p.)\n\nThus, the distance formula is shown to be:\n\n$$d = 35 + \\sqrt{65}\\cos(t + 1.05)$$"}]}],"generatedAt":"2026-06-11T03:05:05.782Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"94c0965b-7c3f-4f9d-a3e1-ff0a9f9683dc","label":"c","content":"Hence, find the minimum distance between the buoy and the dampener plate. Give your answer to the nearest centimetre.","markscheme":"- Identifies that the minimum distance occurs when $\\cos(t + \\alpha) = -1$, giving $d_{min} = 35 - \\sqrt{65}$ M1\n- Calculates $26.9377...$ m and converts to $2694$ cm A1","marks":2,"order":2,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"auxiliary_angle","label":"Auxiliary Angle Method","steps":[{"type":"text","order":0,"title":"Define distance function","content":"The distance $d$ between the buoy and the dampener plate is the difference between their heights:\n\n$$d = h_B - h_D$$ \n\nSubstituting the given expressions for $h_B$ and $h_D$:\n\n$$d = (50 - 7\\sin t) - (15 - 4\\cos t)$$\n$$d = 35 - 7\\sin t + 4\\cos t$$","highlights":[{"text":"h_B = 50 - 7\\sin t","location":"specification"},{"text":"h_D = 15 - 4\\cos t","location":"specification"}]},{"type":"text","order":1,"title":"Apply Auxiliary Angle Method","content":"To easily find the minimum value, we rewrite the trigonometric part, $4\\cos t - 7\\sin t$, in the harmonic form $R\\cos(t + \\alpha)$. \n\nUsing the cosine addition formula:\n\n$$R\\cos(t + \\alpha) = R\\cos t\\cos\\alpha - R\\sin t\\sin\\alpha$$\n\nComparing this to $4\\cos t - 7\\sin t$, we can equate the coefficients:\n\n$$R\\cos\\alpha = 4$$\n$$R\\sin\\alpha = 7$$\n\nTo find the amplitude $R$, we square and add these equations:\n\n$$R^2 = 4^2 + 7^2 = 16 + 49 = 65$$\n$$R = \\sqrt{65}$$"},{"type":"text","order":2,"title":"Express distance harmonically","content":"Using our value of $R = \\sqrt{65}$, we can write the distance function as:\n\n$$d = 35 + \\sqrt{65}\\cos(t + \\alpha)$$\n\nwhere $\\tan\\alpha = \\frac{7}{4}$. Since we only need to find the minimum distance, we do not need to calculate the exact value of the auxiliary angle $\\alpha$."},{"type":"text","order":3,"title":"Find minimum distance","content":"To find the minimum value of $d = 35 + \\sqrt{65}\\cos(t + \\alpha)$, we look at the range of the cosine function. \n\nThe minimum value of $\\cos(t + \\alpha)$ is $-1$.\n\nSubstituting $\\cos(t + \\alpha) = -1$ gives the minimum distance $d_{\\text{min}}$:\n\n$$d_{\\text{min}} = 35 + \\sqrt{65}(-1) = 35 - \\sqrt{65}\\text{ m}$$\n\nThis earns the first mark M1."},{"type":"text","order":4,"title":"Calculate and convert units","content":"Let's calculate the numerical value of $d_{\\text{min}}$:\n\n$$d_{\\text{min}} = 35 - \\sqrt{65} \\approx 26.93774...\\text{ m}$$\n\nThe question asks for the answer to the **nearest centimetre**.\n\nConverting from metres to centimetres by multiplying by $100$:\n\n$$26.93774...\\text{ m} \\times 100 = 2693.774...\\text{ cm}$$\n\nRounding to the nearest whole number gives:\n\n$$2694\\text{ cm}$$\n\nThis final accuracy completes the second mark A1.","calculator":[{"gdc":{"expression":"(35 - √(65)) * 100","instructions":"1. In the run-matrix menu, compute `35 - √(65)`.\n2. Multiply by `100` to convert to cm and round to the nearest integer."},"ti84":{"expression":"(35 - √(65)) * 100","instructions":"1. Enter `35 - √(65)` and press ENTER.\n2. Multiply the result by `100` to convert to centimetres."},"desmos":{"expression":"(35 - \\sqrt{65}) * 100","instructions":"Type the conversion equation to find the exact rounded value."},"description":"Calculate the exact minimum value in metres and convert it to centimetres."}],"highlights":[{"text":"nearest centimetre","location":"specification"}]}]},{"id":"calculus","label":"Calculus Method","steps":[{"type":"text","order":0,"title":"Define the distance function","content":"First, let's write down the expression for the distance, $d(t)$, between the buoy and the dampener plate.\n\nSince the buoy is always higher than the dampener plate (the minimum height of the buoy is $50 - 7 = 43\\text{ m}$, while the maximum height of the dampener plate is $15 + 4 = 19\\text{ m}$), the distance is simply:\n\n$$d(t) = h_B - h_D$$ \n\nSubstituting the expressions for $h_B$ and $h_D$:\n\n$$d(t) = (50 - 7\\sin t) - (15 - 4\\cos t)$$\n$$d(t) = 35 - 7\\sin t + 4\\cos t$$"},{"type":"text","order":1,"title":"Differentiate with respect to $t$","content":"To find the critical points where the minimum distance occurs, we differentiate $d(t)$ with respect to time $t$:\n\n$$\\frac{\\text{d}d}{\\text{d}t} = \\frac{\\text{d}}{\\text{d}t}(35 - 7\\sin t + 4\\cos t)$$\n\nUsing standard derivatives ($(\\sin t)' = \\cos t$ and $(\\cos t)' = -\\sin t$):\n\n$$\\frac{\\text{d}d}{\\text{d}t} = -7\\cos t - 4\\sin t$$"},{"type":"text","order":2,"title":"Find the critical points","content":"We set the derivative equal to zero to find the critical points:\n\n$$-7\\cos t - 4\\sin t = 0$$\n$$-4\\sin t = 7\\cos t$$\n\nDividing both sides by $\\cos t$ (assuming $\\cos t \\neq 0$):\n\n$$\\tan t = -\\frac{7}{4} = -1.75$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\tan t = -\\frac{\\htmlData{anchor=opp}{\\color{@sky}{7}}}{\\htmlData{anchor=adj}{\\color{@amber}{4}}}$"},{"text":"Using a right-angled triangle with opposite side $7$ and adjacent side $4$, the hypotenuse is:"},{"latex":"$$\\text{Hypotenuse} = \\sqrt{\\color{@sky}{7}^2 + \\color{@amber}{4}^2} = \\sqrt{\\htmlData{anchor=hyp}{\\color{@purple}{65}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"opp"},{"color":"amber","style":"text-color","anchor":"adj"}],"connections":[{"to":"hyp","from":"opp","color":"sky","label":null,"style":"dashed"},{"to":"hyp","from":"adj","color":"amber","label":null,"style":"dashed"}]},"order":3,"title":"Identify the minimum distance","description":"We now determine the values of $\\sin t$ and $\\cos t$ that minimize $d(t) = 35 - 7\\sin t + 4\\cos t$."},{"type":"text","order":4,"title":"Evaluate the trigonometric terms","content":"Since $\\tan t = -\\frac{7}{4} < 0$, the angle $t$ lies in either the second or fourth quadrant.\n\nTo minimize $d(t) = 35 - 7\\sin t + 4\\cos t$, we want to make the term $-7\\sin t + 4\\cos t$ as negative as possible.\n\nThis occurs when $\\sin t > 0$ and $\\cos t < 0$ (which places $t$ in the second quadrant):\n\n$$\\sin t = \\frac{7}{\\sqrt{65}}$$\n$$\\cos t = -\\frac{4}{\\sqrt{65}}$$"},{"type":"text","order":5,"title":"Calculate the minimum distance","content":"Substitute these values back into our distance equation:\n\n$$d_{\\text{min}} = 35 - 7\\left(\\frac{7}{\\sqrt{65}}\\right) + 4\\left(-\\frac{4}{\\sqrt{65}}\\right)$$\n$$d_{\\text{min}} = 35 - \\frac{49}{\\sqrt{65}} - \\frac{16}{\\sqrt{65}}$$\n$$d_{\\text{min}} = 35 - \\frac{65}{\\sqrt{65}}$$\n$$d_{\\text{min}} = 35 - \\sqrt{65}$$\n\nUsing a calculator:\n\n$$d_{\\text{min}} \\approx 35 - 8.062257... = 26.9377...\\text{ m}$$"},{"type":"text","order":6,"title":"Convert to nearest centimetre","content":"Since $1\\text{ m} = 100\\text{ cm}$:\n\n$$26.9377...\\text{ m} \\times 100 = 2693.77...\\text{ cm}$$\n\nRounding to the nearest centimetre gives:\n\n$$\\text{Minimum distance} = 2694\\text{ cm}$$\n\nThis confirms the markscheme's result."}]},{"id":"graphical_calculator","label":"Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Define the distance function","content":"The distance $d$ between the buoy and the dampener plate is the difference between their respective heights, $h_B$ and $h_D$:\n\n$$d(t) = h_B - h_D$$\n\nSubstituting the given functions:\n\n$$\\begin{aligned} d(t) &= (50 - 7\\sin t) - (15 - 4\\cos t) \\\\ d(t) &= 35 - 7\\sin t + 4\\cos t \\end{aligned}$$\n\nTo find the minimum distance, we need to find the minimum value of this function.","highlights":[{"text":"height h_B metres of the buoy above the seabed at time t seconds is given by\n\\[h_B = 50 - 7\\sin t\\]","location":"specification"},{"text":"height h_D metres of the dampener plate above the seabed at time t seconds is given by\n\\[h_D = 15 - 4\\cos t\\]","location":"specification"}]},{"type":"text","order":1,"title":"Graph the function","content":"We can plot $y = 35 - 7\\sin x + 4\\cos x$ using a graphing calculator. \n\nEnsure that your calculator is set to **Radian** mode because the time variable $t$ in trigonometric models is measured in radians.","calculator":[{"gdc":{"expression":"35 - 7*sin(x) + 4*cos(x)","instructions":"1. Go to the Graph mode.\n2. Enter the equation: `y1 = 35 - 7*sin(x) + 4*cos(x)`.\n3. Set your view window settings so $x$ goes from $0$ to $2\\pi$ (approx $6.28$) and $y$ goes from $20$ to $50$.\n4. Draw the graph."},"ti84":{"expression":"35 - 7*sin(X) + 4*cos(X)","instructions":"1. Press [MODE] and ensure **RADIAN** is selected.\n2. Press [Y=] and enter `Y1 = 35 - 7*sin(X) + 4*cos(X)`.\n3. Press [WINDOW] and set: `Xmin = 0`, `Xmax = 2*π`, `Ymin = 20`, `Ymax = 50`.\n4. Press [GRAPH] to display the wave-like distance curve."},"desmos":{"expression":"y = 35 - 7\\sin(x) + 4\\cos(x)","instructions":"Type `y = 35 - 7\\sin(x) + 4\\cos(x)` into the input bar. Click on the wrench icon to set the x-axis from `0` to `6.28` and the y-axis from `20` to `50`."},"description":"Plot the distance function on your GDC."}]},{"type":"text","order":2,"title":"Find the minimum value","content":"Now use the minimum finder tool on your graphing calculator to locate the coordinates of the lowest point (the local minimum) on the curve.","calculator":[{"gdc":{"instructions":"1. Press [G-Solv] (usually F5) and select **MIN** (usually F3).\n2. The calculator will automatically jump to the minimum coordinate on the screen and show `x ≈ 4.19` and `y ≈ 26.94`."},"ti84":{"instructions":"1. Press [2nd] [CALC] (above the [TRACE] key) and select `3: minimum`.\n2. Set a **Left Bound** to the left of the lowest point, and press [ENTER].\n3. Set a **Right Bound** to the right of the lowest point, and press [ENTER].\n4. Press [ENTER] again when prompted to **Guess**.\n5. The screen will display the coordinates at the bottom: `X ≈ 4.19` and `Y ≈ 26.94`."},"desmos":{"instructions":"Click directly on the lowest point of the plotted curve. A gray dot will appear. Hover over or click on it to display its coordinates: `(4.189, 26.938)`."},"description":"Locate the coordinates of the minimum point."}]},{"type":"text","order":3,"title":"Convert and round","content":"The minimum value of the distance is the $y$-coordinate of the minimum point:\n\n$$d_{\\min} \\approx 26.9377\\text{ m}$$\n\nSince the question asks for the answer to the nearest centimetre, we convert metres to centimetres by multiplying by $100$:\n\n$$26.9377\\text{ m} \\times 100 \\approx 2693.77\\text{ cm}$$\n\nRounding to the nearest whole centimetre gives:\n\n$$d_{\\min} \\approx 2694\\text{ cm}$$","highlights":[{"text":"Give your answer to the nearest centimetre.","location":"specification"}]}]}],"generatedAt":"2026-06-11T03:05:24.188Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"ae50bae1-e39a-4af8-872b-20077213ce87:individual_evaluation","label":"(a) Individual Evaluation","steps":[{"type":"text","order":0,"title":"Understand \"initial\" height","content":"To find the initial position of any object in a time-dependent model, we look at the start of the time period, which corresponds to $$t = 0$$.\n\nWe will evaluate the heights of the buoy ($h_B$) and the dampener plate ($h_D$) individually at this time.","highlights":[{"text":"initial vertical distance","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$h_B = 50 - 7\\sin(\\htmlData{anchor=t-val-b}{\\color{@sky}{0}})$"},{"latex":"$$h_B = 50 - 7(\\htmlData{anchor=sin-val}{\\color{@sky}{0}})$"},{"latex":"$$h_B = \\htmlData{anchor=hb-final}{\\color{@green}{50}} \\text{ m}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"t-val-b"},{"color":"green","style":"box","anchor":"hb-final"}],"connections":[{"to":"sin-val","from":"t-val-b","color":"sky","label":"sin(0) = 0","style":"solid"}]},"order":1,"title":"Evaluate the buoy height","description":"Substitute $t = 0$ into the equation for the buoy height, $h_B = 50 - 7\\sin t$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$h_D = 15 - 4\\cos(\\htmlData{anchor=t-val-d}{\\color{@sky}{0}})$"},{"latex":"$$h_D = 15 - 4(\\htmlData{anchor=cos-val}{\\color{@sky}{1}})$"},{"latex":"$$h_D = 15 - 4$"},{"latex":"$$h_D = \\htmlData{anchor=hd-final}{\\color{@amber}{11}} \\text{ m}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"t-val-d"},{"color":"amber","style":"box","anchor":"hd-final"}],"connections":[{"to":"cos-val","from":"t-val-d","color":"sky","label":"cos(0) = 1","style":"solid"}]},"order":2,"title":"Evaluate dampener plate height","description":"Now, substitute $t = 0$ into the equation for the dampener plate height, $h_D = 15 - 4\\cos t$."},{"type":"text","order":3,"title":"Calculate the distance","content":"The vertical distance between the buoy and the dampener plate is the difference between their respective heights:\n\n$$\\text{Vertical Distance} = h_B - h_D$$\n\nSubstituting our calculated values:\n\n$$\\text{Vertical Distance} = 50 - 11 = 39 \\text{ m}$$\n\nThis confirms the initial vertical distance is indeed **$39$ metres**.","highlights":[{"text":"39","location":"specification"}]}]},{"id":"ae50bae1-e39a-4af8-872b-20077213ce87:distance_function","label":"(a) Distance Function Formulation","steps":[{"type":"text","order":0,"title":"Define the distance function","content":"To find the vertical distance between the buoy and the dampener plate at any time $t$, we can define a single distance function, $d(t)$, representing the difference between their heights:\n\n$$d(t) = h_B(t) - h_D(t)$$\n\nSubstituting the given expressions for $h_B$ and $h_D$:\n\n$$d(t) = (50 - 7\\sin t) - (15 - 4\\cos t)$$\n\nThis single function gives the vertical distance between the two components at any time $t$ in seconds.","highlights":[{"text":"h_B = 50 - 7\\sin t","location":"specification"},{"text":"h_D = 15 - 4\\cos t","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$d(t) = 50 - 7\\sin t - 15 + 4\\cos t$"},{"latex":"$$d(t) = \\htmlData{anchor=const}{\\color{@sky}{35}} - 7\\sin t + 4\\cos t$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"const"}],"connections":[]},"order":1,"title":"Simplify the function","description":"We simplify the expression by combining the constant numerical terms: \n\n$$50 - 15 = 35$$\n\nThis gives us the simplified distance function:\n\n$$d(t) = 35 - 7\\sin t + 4\\cos t$$"},{"type":"text","order":2,"title":"Find the initial distance","content":"The \"initial\" distance refers to the distance at the very start of the operating cycle, which is when time $t = 0$.\n\nSubstitute $t = 0$ into our simplified distance function $d(t)$:\n\n$$d(0) = 35 - 7\\sin(0) + 4\\cos(0)$$\n\nRecall the exact values for the trigonometric functions at $0$:\n- $\\sin(0) = 0$\n- $\\cos(0) = 1$\n\nNow substitute these values in:\n\n$$\\begin{aligned} d(0) &= 35 - 7(0) + 4(1) \\\\ d(0) &= 35 + 4 \\\\ d(0) &= 39 \\text{ metres} \\end{aligned}$$\n\nThis matches the expected answer and earns both marks under the markscheme."}]},{"id":"2dd37b01-45d7-4392-b31d-eb55a13b7302:compound_angle_expansion","label":"(b) Compound Angle Expansion","steps":[{"type":"text","order":0,"title":"Subtract the height expressions","content":"To find the distance $d$ between the buoy and the dampener plate at time $t$, we subtract the expression for the height of the dampener plate $h_D$ from the height of the buoy $h_B$:\n\n$$d = h_B - h_D$$\n\nSubstituting our given expressions:\n\n$$d = (50 - 7\\sin t) - (15 - 4\\cos t)$$\n$$d = 50 - 15 + 4\\cos t - 7\\sin t$$\n$$d = 35 + 4\\cos t - 7\\sin t$$\n\nThis earns the first mark **(B1)**.","highlights":[{"text":"h_B = 50 - 7\\sin t","location":"specification"},{"text":"h_D = 15 - 4\\cos t","location":"specification"}]},{"type":"text","order":1,"title":"Use compound angle expansion","content":"We need to express $4\\cos t - 7\\sin t$ in the form $R\\cos(t + \\alpha)$.\n\nUsing the compound angle identity for cosine:\n\n$$\\cos(A + B) = \\cos A \\cos B - \\sin A \\sin B$$\n\nExpanding $R\\cos(t + \\alpha)$ gives us:\n\n$$R\\cos(t + \\alpha) = R\\cos t \\cos \\alpha - R\\sin t \\sin \\alpha$$\n\nWe now equate this expansion to our expression $4\\cos t - 7\\sin t$:\n\n$$R\\cos t \\cos \\alpha - R\\sin t \\sin \\alpha = 4\\cos t - 7\\sin t$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(\\htmlData{anchor=lh1}{\\color{@sky}{R \\cos \\alpha}})\\cos t - (\\htmlData{anchor=lh2}{\\color{@amber}{R \\sin \\alpha}})\\sin t = \\htmlData{anchor=rh1}{\\color{@sky}{4}}\\cos t - \\htmlData{anchor=rh2}{\\color{@amber}{7}}\\sin t$"},{"latex":"$$$R \\cos \\alpha = 4 \\quad \\text{and} \\quad R \\sin \\alpha = 7$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"lh1"},{"color":"sky","style":"text-color","anchor":"rh1"},{"color":"amber","style":"text-color","anchor":"lh2"},{"color":"amber","style":"text-color","anchor":"rh2"}],"connections":[{"to":"rh1","from":"lh1","color":"sky","label":"equate cos","style":"solid"},{"to":"rh2","from":"lh2","color":"amber","label":"equate sin","style":"solid"}]},"order":2,"title":"Equate the coefficients","description":"By comparing the coefficients of $\\cos t$ and $\\sin t$ on both sides of the equation, we get two simultaneous equations."},{"type":"text","order":3,"title":"Solve for R","content":"To find $R$, we square and add our two equations:\n\n$$(R \\sin \\alpha)^2 + (R \\cos \\alpha)^2 = 7^2 + 4^2$$\n$$R^2 (\\sin^2 \\alpha + \\cos^2 \\alpha) = 49 + 16$$\n\nSince $\\sin^2 \\alpha + \\cos^2 \\alpha = 1$, we get:\n\n$$R^2 = 65$$\n$$R = \\sqrt{65} \\approx 8.06$$\n\nThis step earns the second mark **(M1)**."},{"type":"text","order":4,"title":"Solve for alpha","content":"To find $\\alpha$, we divide the sine equation by the cosine equation:\n\n$$\\frac{R \\sin \\alpha}{R \\cos \\alpha} = \\frac{7}{4}$$\n$$\\tan \\alpha = \\frac{7}{4}$$\n\nThis earns the third mark **(M1)**.\n\nNow we calculate $\\alpha$ in radians:\n\n$$\\alpha = \\arctan\\left(\\frac{7}{4}\\right) \\approx 1.05166... \\approx 1.05 \\text{ rad}$$\n\nRounding to two decimal places gives $\\alpha = 1.05$ **(A1)**.","calculator":[{"gdc":{"expression":"arctan(7/4)","instructions":"1. Check that the angle settings are set to Radians.\n2. Access the inverse tangent function: press [SHIFT] [TAN] or locate 'arctan'.\n3. Input 7/4 and execute."},"ti84":{"expression":"tan⁻¹(7/4)","instructions":"1. Ensure your calculator is in Radian mode by pressing [MODE], highlighting 'RADIAN', and pressing [ENTER].\n2. Press [2nd] [TAN] to select $\\tan^{-1}$.\n3. Type 7 / 4 and press [ENTER]."},"desmos":{"expression":"\\arctan\\left(\\frac{7}{4}\\right)","instructions":"1. Set the grid to 'Radians' mode in the Graph Settings (wrench icon).\n2. Type 'arctan(7/4)' in the input bar to get the result."},"description":"Calculate the angle $\\alpha$ in radians using your calculator."}]},{"type":"text","order":5,"title":"State the final expression","content":"Now we substitute the values of $R$ and $\\alpha$ back into our expression for the distance $d$:\n\n$$d = 35 + \\sqrt{65}\\cos(t + 1.05)$$\n\nwhere $R = \\sqrt{65}$ (or $\\approx 8.06$) and $\\alpha = 1.05$ (to $2$ decimal places)."}]},{"id":"2dd37b01-45d7-4392-b31d-eb55a13b7302:direct_formula","label":"(b) Direct R-Formula Application","steps":[{"type":"text","order":0,"title":"Set up distance equation","content":"The distance $d$ between the floating buoy and the submerged dampener plate is found by subtracting the height of the plate, $h_D$, from the height of the buoy, $h_B$:\n\n$$d = h_B - h_D$$ \n\nSubstituting the given expressions:\n\n$$d = (50 - 7\\sin t) - (15 - 4\\cos t)$$\n\n$$d = 50 - 7\\sin t - 15 + 4\\cos t$$\n\n$$d = 35 + 4\\cos t - 7\\sin t$$","highlights":[{"text":"h_B = 50 - 7\\sin t","location":"specification"},{"text":"h_D = 15 - 4\\cos t","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{We compare } \\htmlData{anchor=expr}{\\color{@sky}{4\\cos t - 7\\sin t}} \\text{ to the general R-formula format:}$"},{"latex":"$$$a\\cos t - b\\sin t = R\\cos(t + \\alpha)$$"},{"latex":"$$\\text{By matching coefficients, we identify:}$"},{"latex":"$$$\\htmlData{anchor=a-val}{\\color{@purple}{a = 4}} \\quad \\text{and} \\quad \\htmlData{anchor=b-val}{\\color{@teal}{b = 7}}$$"}],"highlights":[{"color":"sky","style":"box","anchor":"expr"},{"color":"purple","style":"text-color","anchor":"a-val"},{"color":"teal","style":"text-color","anchor":"b-val"}],"connections":[{"to":"a-val","from":"expr","color":"purple","label":"cosine coefficient","style":"solid"},{"to":"b-val","from":"expr","color":"teal","label":"sine coefficient","style":"solid"}]},"order":1,"title":"Apply the R-formula","description":"We match the coefficients of our equation to the standard R-formula."},{"type":"text","order":2,"title":"Calculate amplitude R","content":"The constant $R$ represents the amplitude of the combined trigonometric function and is given directly by:\n\n$$R = \\sqrt{a^2 + b^2}$$\n\nSubstituting $a = 4$ and $b = 7$:\n\n$$R = \\sqrt{4^2 + 7^2}$$\n\n$$R = \\sqrt{16 + 49} = \\sqrt{65} \\approx 8.06$$"},{"type":"text","order":3,"title":"Calculate phase angle alpha","content":"The phase angle $\\alpha$ is determined directly using the relation:\n\n$$\\tan\\alpha = \\frac{b}{a}$$\n\nSubstituting our values:\n\n$$\\tan\\alpha = \\frac{7}{4}$$\n\n$$\\alpha = \\arctan\\left(\\frac{7}{4}\\right)$$\n\nUsing a calculator in radian mode:\n\n$$\\alpha \\approx 1.05165... \\text{ rad}$$ \n\nRounding to two decimal places, we get:\n\n$$\\alpha \\approx 1.05$$","calculator":[{"gdc":{"expression":"arctan(7/4)","instructions":"Set angle mode to Radians. Navigate to the run-matrix screen and calculate the inverse tangent of 7/4."},"ti84":{"expression":"tan⁻¹(7/4)","instructions":"Ensure your calculator is in RADIAN mode (Press [MODE], select 'RADIAN', and press [ENTER]). Then press [2nd] [TAN] ( 7 / 4 ) [ENTER]."},"desmos":{"expression":"\\arctan(7/4)","instructions":"Ensure Desmos is set to Radians. Type 'arctan(7/4)' or use the keypad to find the inverse tangent."},"description":"Calculate the inverse tangent of 1.75 to find the angle in radians."}]},{"type":"text","order":4,"title":"Write final expression","content":"Now we substitute $R$ and $\\alpha$ back into our overall expression for the distance $d$:\n\n$$d = 35 + R\\cos(t + \\alpha)$$\n\nWhere:\n- $R = \\sqrt{65}$ (or $\\approx 8.06$)\n- $\\alpha = 1.05$ (to $2$ d.p.)\n\nThus, the distance formula is shown to be:\n\n$$d = 35 + \\sqrt{65}\\cos(t + 1.05)$$"}]},{"id":"94c0965b-7c3f-4f9d-a3e1-ff0a9f9683dc:auxiliary_angle","label":"(c) Auxiliary Angle Method","steps":[{"type":"text","order":0,"title":"Define distance function","content":"The distance $d$ between the buoy and the dampener plate is the difference between their heights:\n\n$$d = h_B - h_D$$ \n\nSubstituting the given expressions for $h_B$ and $h_D$:\n\n$$d = (50 - 7\\sin t) - (15 - 4\\cos t)$$\n$$d = 35 - 7\\sin t + 4\\cos t$$","highlights":[{"text":"h_B = 50 - 7\\sin t","location":"specification"},{"text":"h_D = 15 - 4\\cos t","location":"specification"}]},{"type":"text","order":1,"title":"Apply Auxiliary Angle Method","content":"To easily find the minimum value, we rewrite the trigonometric part, $4\\cos t - 7\\sin t$, in the harmonic form $R\\cos(t + \\alpha)$. \n\nUsing the cosine addition formula:\n\n$$R\\cos(t + \\alpha) = R\\cos t\\cos\\alpha - R\\sin t\\sin\\alpha$$\n\nComparing this to $4\\cos t - 7\\sin t$, we can equate the coefficients:\n\n$$R\\cos\\alpha = 4$$\n$$R\\sin\\alpha = 7$$\n\nTo find the amplitude $R$, we square and add these equations:\n\n$$R^2 = 4^2 + 7^2 = 16 + 49 = 65$$\n$$R = \\sqrt{65}$$"},{"type":"text","order":2,"title":"Express distance harmonically","content":"Using our value of $R = \\sqrt{65}$, we can write the distance function as:\n\n$$d = 35 + \\sqrt{65}\\cos(t + \\alpha)$$\n\nwhere $\\tan\\alpha = \\frac{7}{4}$. Since we only need to find the minimum distance, we do not need to calculate the exact value of the auxiliary angle $\\alpha$."},{"type":"text","order":3,"title":"Find minimum distance","content":"To find the minimum value of $d = 35 + \\sqrt{65}\\cos(t + \\alpha)$, we look at the range of the cosine function. \n\nThe minimum value of $\\cos(t + \\alpha)$ is $-1$.\n\nSubstituting $\\cos(t + \\alpha) = -1$ gives the minimum distance $d_{\\text{min}}$:\n\n$$d_{\\text{min}} = 35 + \\sqrt{65}(-1) = 35 - \\sqrt{65}\\text{ m}$$\n\nThis earns the first mark M1."},{"type":"text","order":4,"title":"Calculate and convert units","content":"Let's calculate the numerical value of $d_{\\text{min}}$:\n\n$$d_{\\text{min}} = 35 - \\sqrt{65} \\approx 26.93774...\\text{ m}$$\n\nThe question asks for the answer to the **nearest centimetre**.\n\nConverting from metres to centimetres by multiplying by $100$:\n\n$$26.93774...\\text{ m} \\times 100 = 2693.774...\\text{ cm}$$\n\nRounding to the nearest whole number gives:\n\n$$2694\\text{ cm}$$\n\nThis final accuracy completes the second mark A1.","calculator":[{"gdc":{"expression":"(35 - √(65)) * 100","instructions":"1. In the run-matrix menu, compute `35 - √(65)`.\n2. Multiply by `100` to convert to cm and round to the nearest integer."},"ti84":{"expression":"(35 - √(65)) * 100","instructions":"1. Enter `35 - √(65)` and press ENTER.\n2. Multiply the result by `100` to convert to centimetres."},"desmos":{"expression":"(35 - \\sqrt{65}) * 100","instructions":"Type the conversion equation to find the exact rounded value."},"description":"Calculate the exact minimum value in metres and convert it to centimetres."}],"highlights":[{"text":"nearest centimetre","location":"specification"}]}]},{"id":"94c0965b-7c3f-4f9d-a3e1-ff0a9f9683dc:calculus","label":"(c) Calculus Method","steps":[{"type":"text","order":0,"title":"Define the distance function","content":"First, let's write down the expression for the distance, $d(t)$, between the buoy and the dampener plate.\n\nSince the buoy is always higher than the dampener plate (the minimum height of the buoy is $50 - 7 = 43\\text{ m}$, while the maximum height of the dampener plate is $15 + 4 = 19\\text{ m}$), the distance is simply:\n\n$$d(t) = h_B - h_D$$ \n\nSubstituting the expressions for $h_B$ and $h_D$:\n\n$$d(t) = (50 - 7\\sin t) - (15 - 4\\cos t)$$\n$$d(t) = 35 - 7\\sin t + 4\\cos t$$"},{"type":"text","order":1,"title":"Differentiate with respect to $t$","content":"To find the critical points where the minimum distance occurs, we differentiate $d(t)$ with respect to time $t$:\n\n$$\\frac{\\text{d}d}{\\text{d}t} = \\frac{\\text{d}}{\\text{d}t}(35 - 7\\sin t + 4\\cos t)$$\n\nUsing standard derivatives ($(\\sin t)' = \\cos t$ and $(\\cos t)' = -\\sin t$):\n\n$$\\frac{\\text{d}d}{\\text{d}t} = -7\\cos t - 4\\sin t$$"},{"type":"text","order":2,"title":"Find the critical points","content":"We set the derivative equal to zero to find the critical points:\n\n$$-7\\cos t - 4\\sin t = 0$$\n$$-4\\sin t = 7\\cos t$$\n\nDividing both sides by $\\cos t$ (assuming $\\cos t \\neq 0$):\n\n$$\\tan t = -\\frac{7}{4} = -1.75$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\tan t = -\\frac{\\htmlData{anchor=opp}{\\color{@sky}{7}}}{\\htmlData{anchor=adj}{\\color{@amber}{4}}}$"},{"text":"Using a right-angled triangle with opposite side $7$ and adjacent side $4$, the hypotenuse is:"},{"latex":"$$\\text{Hypotenuse} = \\sqrt{\\color{@sky}{7}^2 + \\color{@amber}{4}^2} = \\sqrt{\\htmlData{anchor=hyp}{\\color{@purple}{65}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"opp"},{"color":"amber","style":"text-color","anchor":"adj"}],"connections":[{"to":"hyp","from":"opp","color":"sky","label":null,"style":"dashed"},{"to":"hyp","from":"adj","color":"amber","label":null,"style":"dashed"}]},"order":3,"title":"Identify the minimum distance","description":"We now determine the values of $\\sin t$ and $\\cos t$ that minimize $d(t) = 35 - 7\\sin t + 4\\cos t$."},{"type":"text","order":4,"title":"Evaluate the trigonometric terms","content":"Since $\\tan t = -\\frac{7}{4} < 0$, the angle $t$ lies in either the second or fourth quadrant.\n\nTo minimize $d(t) = 35 - 7\\sin t + 4\\cos t$, we want to make the term $-7\\sin t + 4\\cos t$ as negative as possible.\n\nThis occurs when $\\sin t > 0$ and $\\cos t < 0$ (which places $t$ in the second quadrant):\n\n$$\\sin t = \\frac{7}{\\sqrt{65}}$$\n$$\\cos t = -\\frac{4}{\\sqrt{65}}$$"},{"type":"text","order":5,"title":"Calculate the minimum distance","content":"Substitute these values back into our distance equation:\n\n$$d_{\\text{min}} = 35 - 7\\left(\\frac{7}{\\sqrt{65}}\\right) + 4\\left(-\\frac{4}{\\sqrt{65}}\\right)$$\n$$d_{\\text{min}} = 35 - \\frac{49}{\\sqrt{65}} - \\frac{16}{\\sqrt{65}}$$\n$$d_{\\text{min}} = 35 - \\frac{65}{\\sqrt{65}}$$\n$$d_{\\text{min}} = 35 - \\sqrt{65}$$\n\nUsing a calculator:\n\n$$d_{\\text{min}} \\approx 35 - 8.062257... = 26.9377...\\text{ m}$$"},{"type":"text","order":6,"title":"Convert to nearest centimetre","content":"Since $1\\text{ m} = 100\\text{ cm}$:\n\n$$26.9377...\\text{ m} \\times 100 = 2693.77...\\text{ cm}$$\n\nRounding to the nearest centimetre gives:\n\n$$\\text{Minimum distance} = 2694\\text{ cm}$$\n\nThis confirms the markscheme's result."}]},{"id":"94c0965b-7c3f-4f9d-a3e1-ff0a9f9683dc:graphical_calculator","label":"(c) Using Graphing Calculator","steps":[{"type":"text","order":0,"title":"Define the distance function","content":"The distance $d$ between the buoy and the dampener plate is the difference between their respective heights, $h_B$ and $h_D$:\n\n$$d(t) = h_B - h_D$$\n\nSubstituting the given functions:\n\n$$\\begin{aligned} d(t) &= (50 - 7\\sin t) - (15 - 4\\cos t) \\\\ d(t) &= 35 - 7\\sin t + 4\\cos t \\end{aligned}$$\n\nTo find the minimum distance, we need to find the minimum value of this function.","highlights":[{"text":"height h_B metres of the buoy above the seabed at time t seconds is given by\n\\[h_B = 50 - 7\\sin t\\]","location":"specification"},{"text":"height h_D metres of the dampener plate above the seabed at time t seconds is given by\n\\[h_D = 15 - 4\\cos t\\]","location":"specification"}]},{"type":"text","order":1,"title":"Graph the function","content":"We can plot $y = 35 - 7\\sin x + 4\\cos x$ using a graphing calculator. \n\nEnsure that your calculator is set to **Radian** mode because the time variable $t$ in trigonometric models is measured in radians.","calculator":[{"gdc":{"expression":"35 - 7*sin(x) + 4*cos(x)","instructions":"1. Go to the Graph mode.\n2. Enter the equation: `y1 = 35 - 7*sin(x) + 4*cos(x)`.\n3. Set your view window settings so $x$ goes from $0$ to $2\\pi$ (approx $6.28$) and $y$ goes from $20$ to $50$.\n4. Draw the graph."},"ti84":{"expression":"35 - 7*sin(X) + 4*cos(X)","instructions":"1. Press [MODE] and ensure **RADIAN** is selected.\n2. Press [Y=] and enter `Y1 = 35 - 7*sin(X) + 4*cos(X)`.\n3. Press [WINDOW] and set: `Xmin = 0`, `Xmax = 2*π`, `Ymin = 20`, `Ymax = 50`.\n4. Press [GRAPH] to display the wave-like distance curve."},"desmos":{"expression":"y = 35 - 7\\sin(x) + 4\\cos(x)","instructions":"Type `y = 35 - 7\\sin(x) + 4\\cos(x)` into the input bar. Click on the wrench icon to set the x-axis from `0` to `6.28` and the y-axis from `20` to `50`."},"description":"Plot the distance function on your GDC."}]},{"type":"text","order":2,"title":"Find the minimum value","content":"Now use the minimum finder tool on your graphing calculator to locate the coordinates of the lowest point (the local minimum) on the curve.","calculator":[{"gdc":{"instructions":"1. Press [G-Solv] (usually F5) and select **MIN** (usually F3).\n2. The calculator will automatically jump to the minimum coordinate on the screen and show `x ≈ 4.19` and `y ≈ 26.94`."},"ti84":{"instructions":"1. Press [2nd] [CALC] (above the [TRACE] key) and select `3: minimum`.\n2. Set a **Left Bound** to the left of the lowest point, and press [ENTER].\n3. Set a **Right Bound** to the right of the lowest point, and press [ENTER].\n4. Press [ENTER] again when prompted to **Guess**.\n5. The screen will display the coordinates at the bottom: `X ≈ 4.19` and `Y ≈ 26.94`."},"desmos":{"instructions":"Click directly on the lowest point of the plotted curve. A gray dot will appear. Hover over or click on it to display its coordinates: `(4.189, 26.938)`."},"description":"Locate the coordinates of the minimum point."}]},{"type":"text","order":3,"title":"Convert and round","content":"The minimum value of the distance is the $y$-coordinate of the minimum point:\n\n$$d_{\\min} \\approx 26.9377\\text{ m}$$\n\nSince the question asks for the answer to the nearest centimetre, we convert metres to centimetres by multiplying by $100$:\n\n$$26.9377\\text{ m} \\times 100 \\approx 2693.77\\text{ cm}$$\n\nRounding to the nearest whole centimetre gives:\n\n$$d_{\\min} \\approx 2694\\text{ cm}$$","highlights":[{"text":"Give your answer to the nearest centimetre.","location":"specification"}]}]}],"generatedAt":"2026-06-11T03:05:24.296Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"f85f2ec3-1bdf-4288-bfe5-f01eb0496eb4","revisionId":"01e46466-326c-406f-84b6-2da1605a5648","questionNumber":null,"questionType":"LA","specification":"Solve, for $0° < x < 360°$, the equation\n\n$\\operatorname{cosec}^2 x - 2\\cot x = 4$","difficultyBand":"medium","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"8435fa23-ecb8-4b32-8306-f956cd521b40","label":null,"content":"Give your answers to one decimal place where appropriate.","markscheme":"- Uses $\\operatorname{cosec}^2 x \\equiv 1 + \\cot^2 x$ M1\n- $\\cot^2 x - 2\\cot x - 3 = 0$ A1\n- $(\\cot x - 3)(\\cot x + 1) = 0$ M1\n- $\\cot x = 3$ gives $\\tan x = \\frac{1}{3}$, so $x = 18.4°$ and $x = 198.4°$ A1\n- $\\cot x = -1$ gives $\\tan x = -1$ M1\n- $x = 135°$ and $x = 315°$ A1\n\nAll four values are required.","marks":6,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"fe99504b-989a-4428-a3dc-b0b213a6ab78","revisionId":"470a60ac-22b9-454a-9fab-4d8b661c3f67","questionNumber":null,"questionType":"LA","specification":"Figure 1 below shows the shape $ABCD$. The sector $ABC$ is a sector of a circle with centre $A$ and radius 7 cm, and the angle $BAC$ is 0.6 radians. The angle $BAD$ is a right angle, $AB = 7$ cm and $AD = 5$ cm.\n\n![Figure for question 8](https://assets.mathsgenie.co.uk/booklet/2da0ecf6142879aeeb1a3c4a2b6b0fa60b5d9dd2711f8aba2f0e4317dabcf29e.svg)","difficultyBand":"medium","tier":null,"totalMarks":7,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"0ef30251-080f-489f-911b-0bb1f9daf1d9","label":"a","content":"Find the length of the arc $BC$.","markscheme":"- $7 \\times 0.6 = 4.2$ cm B1","marks":1,"order":0,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"93c93d6f-507e-4b2f-8830-56e1aeb27d04","label":"b","content":"Calculate the area of the sector $ABC$.","markscheme":"- $\\frac{1}{2} \\times 7^2 \\times 0.6$ M1\n- $= 14.7$ cm$^2$ A1","marks":2,"order":1,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"1a6a24a3-862b-4fec-9365-30e3a035e8c9","label":"c","content":"Find the size of the angle $CAD$, in radians.","markscheme":"- $\\frac{\\pi}{2} - 0.6 = 0.971$ radians B1","marks":1,"order":2,"markingRubric":null,"tableImages":[],"explanation":null,"options":[]},{"id":"bc3d0040-7a7b-4408-801f-4ac811a8abdb","label":"d","content":"Find the total area of the shape $ABCD$, giving your answer to three significant figures.","markscheme":"- Area of triangle $ACD = \\frac{1}{2} \\times 7 \\times 5 \\times \\sin(0.9708)$ M1\n- $= 14.44$ cm$^2$ A1\n- Total area $= 14.7 + 14.44 = 29.1$ cm$^2$ A1\n\n$\\sin\\left(\\frac{\\pi}{2} - 0.6\\right) = \\cos 0.6 = 0.82534$, so the triangle area may be written $17.5\\cos 0.6$ 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1.8 E: Trigonometry

Question 167

1.8 E: Trigonometry Questions

1.8 E: Trigonometry

Question 167

1.8 E: Trigonometry Questions