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angle in its specified inverse-trigonometric range. Other solutions are found using the graph’s symmetry and periodicity."},{"text":"The reference angle.","correct":false,"explanation":"A reference angle is the acute angle made with the x-axis, not the first calculator output. The first angle solution provided by a calculator is called the principal value."},{"text":"The general solution.","correct":false,"explanation":"The general solution includes every angle satisfying the equation, usually expressed using a period. A calculator initially gives only one principal value, not the complete general solution."},{"text":"The critical angle.","correct":false,"explanation":"A critical angle is not the standard term for an inverse-trigonometric calculator result. The first angle returned by the calculator is the principal value."}]},{"id":"30e25b7e-3436-431e-8469-eb1102a49100","cardType":"cloze","points":null,"front":"The period of sine and cosine is $\\text{[...]}$, whereas the period of tangent is $180^\\circ$.","back":"The period of sine and cosine is $\\boxed{360^\\circ}$, whereas the period of tangent is $180^\\circ$.","mcqOptions":[{"text":"$$360^\\circ$","correct":true,"explanation":"Sine and cosine repeat after one full revolution, which is $360^\\circ$. Tangent repeats after $180^\\circ$ because its values recur after half a revolution."},{"text":"$$180^\\circ$","correct":false,"explanation":"$$180^\\circ$ is the period of tangent, not sine and cosine. Sine and cosine need a full revolution before their values repeat, so their period is $360^\\circ$."},{"text":"$$90^\\circ$","correct":false,"explanation":"Although sine and cosine have special values at $90^\\circ$ intervals, their complete cycles do not repeat then. Their period is $360^\\circ$, while tangent’s is $180^\\circ$."},{"text":"$$720^\\circ$","correct":false,"explanation":"$$720^\\circ$ represents two complete revolutions, so it is a multiple of the period rather than the fundamental period. Sine and cosine first repeat after $360^\\circ$."}]},{"id":"7263d3ea-e38c-4946-a277-10a9232c812d","cardType":"cloze","points":null,"front":"The period of sine and cosine is $360^\\circ$, whereas the period of tangent is $\\text{[...]}$.","back":"The period of sine and cosine is $360^\\circ$, whereas the period of tangent is $\\boxed{180^\\circ}$.","mcqOptions":[{"text":"$$180^\\circ$","correct":true,"explanation":"Tangent repeats after half a revolution because both sine and cosine change sign, leaving their ratio unchanged. Therefore its period is $180^\\circ$."},{"text":"$$360^\\circ$","correct":false,"explanation":"$$360^\\circ$ is the period of sine and cosine, not tangent. Tangent’s ratio repeats after $180^\\circ$, when both numerator and denominator have changed sign."},{"text":"$$90^\\circ$","correct":false,"explanation":"Tangent has notable values at $90^\\circ$ intervals, but it is undefined at odd multiples of $90^\\circ$. Its repeating cycle is $180^\\circ$."},{"text":"$$720^\\circ$","correct":false,"explanation":"$$720^\\circ$ is two complete tangent cycles, not the smallest repeating interval. The fundamental period of tangent is $180^\\circ$."}]},{"id":"9fe08b7e-02c9-4be8-8433-27a3e8c72c42","cardType":"basic","points":2,"front":"When solving $\\cos(2x+10^\\circ)=0.4$ for $0 \\le x < 180^\\circ$, what is the first step regarding the interval?","back":"Transform the interval for the inside angle: $10^\\circ \\le (2x+10^\\circ) < 370^\\circ$.","mcqOptions":[{"text":"Transform the interval for the inside angle: $10^\\circ \\le (2x+10^\\circ) < 370^\\circ$.","correct":true,"explanation":"The inside angle is $2x+10^\\circ$. Applying this expression to $0\\le x<180^\\circ$ gives $10^\\circ\\le 2x+10^\\circ<370^\\circ$, the correct interval for solving."},{"text":"Transform the interval for the inside angle: $10^\\circ le (2x+10^\\circ) < 360^\\circ$.","correct":false,"explanation":"The upper endpoint must come from $2(180^\\circ)+10^\\circ=370^\\circ$, not $360^\\circ$. Forgetting the added $10^\\circ$ at the upper end gives an interval that is too short."},{"text":"Transform the interval for the inside angle: $0^\\circ le (2x+10^\\circ) < 180^\\circ$.","correct":false,"explanation":"This leaves the original interval unchanged instead of applying the inside expression. Multiplying by $2$ and adding $10^\\circ$ gives $10^\\circ\\le 2x+10^\\circ<370^\\circ$."},{"text":"Transform the interval for the inside angle: $10^\\circ le (2x+10^\\circ) < 190^\\circ$.","correct":false,"explanation":"The upper endpoint incorrectly adds $10^\\circ$ to $180^\\circ$ but ignores the factor of $2$. The correct upper endpoint is $2(180^\\circ)+10^\\circ=370^\\circ$."}]},{"id":"830ec6de-0f6f-40e5-ae68-993c6d432d83","cardType":"cloze","points":null,"front":"For real angles, the values of $\\sin x$ and $\\cos x$ must always lie between $\\text{[...]}$ and $1$.","back":"For real angles, the values of $\\sin x$ and $\\cos x$ must always lie between $\\boxed{-1}$ and $1$.","mcqOptions":[{"text":"$$-1$","correct":true,"explanation":"For every real angle, sine and cosine cannot be less than $-1$. Their ranges are both from $-1$ to $1$, so the lower bound is $-1$."},{"text":"$$0$","correct":false,"explanation":"Zero is within the range of sine and cosine, but it is not their lower bound. Both functions can take negative values down to $-1$, so the correct lower bound is $-1$."},{"text":"$$-2$","correct":false,"explanation":"Sine and cosine never reach values below $-1$; their outputs are restricted to the interval $[-1,1]$. The lower bound is therefore $-1$, not $-2$."},{"text":"$$1$","correct":false,"explanation":"One is the upper bound of both sine and cosine, not the lower bound. Their values lie between $-1$ and $1$, so the missing lower value is $-1$."}]},{"id":"593766b2-de4a-4ae2-b0df-f170273060dc","cardType":"cloze","points":null,"front":"For real angles, the values of $\\sin x$ and $\\cos x$ must always lie between $-1$ and $\\text{[...]}$.","back":"For real angles, the values of $\\sin x$ and $\\cos x$ must always lie between $-1$ and $\\boxed{1}$.","mcqOptions":[{"text":"$$1$","correct":true,"explanation":"Sine and cosine have maximum values of $1$ for real angles. Together with their minimum value of $-1$, this gives the range $[-1,1]$."},{"text":"$$0$","correct":false,"explanation":"Zero is a value that sine and cosine can take, but it is not their maximum. Both functions can reach $1$, so the correct upper bound is $1$."},{"text":"$$-1$","correct":false,"explanation":"$$-1$ is the minimum value of sine and cosine, not the maximum. Their values extend upward to $1$, which is the required upper bound."},{"text":"$$2$","correct":false,"explanation":"Real sine and cosine values cannot exceed $1$. The range is $[-1,1]$, so the correct upper endpoint is $1$, not $2$."}]},{"id":"91610b5e-599f-4733-9c12-8f8b83342cad","cardType":"basic","points":1,"front":"What must you do when taking the square root to solve an equation like $\\tan^2 x = 5$?","back":"Account for both the positive and negative roots: $\\tan x = \\pm\\sqrt{5}$.","mcqOptions":[{"text":"Account for both the positive and negative roots: $\\tan x = \\pm\\sqrt{5}$.","correct":true,"explanation":"Taking a square root gives two possibilities because both $(\\sqrt{5})^2$ and $(-\\sqrt{5})^2$ equal $5$. Therefore $\\tan x=\\pm\\sqrt{5}$ must be considered."},{"text":"Account for only the positive root: $\\tan x = \\sqrt{5}$.","correct":false,"explanation":"Keeping only the positive root loses solutions where tangent is negative. Since squaring removes the sign, both possibilities must be included: $\\tan x=\\pm\\sqrt{5}$."},{"text":"Account for only the negative root: $\\tan x = -\\sqrt{5}$.","correct":false,"explanation":"Choosing only the negative root discards solutions where tangent is positive. The equation has two square-root possibilities, so the correct result is $\\tan x=\\pm\\sqrt{5}$."},{"text":"Account for both the positive and negative roots: $\\tan x = \\pm 5$.","correct":false,"explanation":"The square root of $5$ is $\\sqrt{5}$, not $5$. The sign must also be retained, giving $\\tan x=\\pm\\sqrt{5}$ rather than $\\pm5$."}]},{"id":"8b38bdb7-0697-4b4e-92ca-a9b051cfa9af","cardType":"cloze","points":null,"front":"The trigonometric identity used to convert $\\sin^2 x$ into terms of cosine is $\\text{[...]}$.","back":"The trigonometric identity used to convert $\\sin^2 x$ into terms of cosine is $\\boxed{\\sin^2 x = 1 - \\cos^2 x}$.","mcqOptions":[{"text":"$$\\sin^2 x = 1 - \\cos^2 x$","correct":true,"explanation":"Rearranging the Pythagorean identity $\\sin^2x+\\cos^2x=1$ gives $\\sin^2x=1-\\cos^2x$, directly expressing the sine term using cosine."},{"text":"$$\\sin^2 x = 1 + \\cos^2 x$","correct":false,"explanation":"The Pythagorean identity requires the squared terms to add to $1$, so rearranging introduces subtraction, not addition. The correct identity is $\\sin^2x=1-\\cos^2x$."},{"text":"$$\\sin^2 x = \\cos^2 x - 1$","correct":false,"explanation":"Rearranging incorrectly places the constant and cosine term in the wrong order, producing values that cannot represent $\\sin^2x$. The correct form is $1-\\cos^2x$."},{"text":"$$\\sin^2 x = 1 - \\cos x$","correct":false,"explanation":"The cosine term must also be squared in the Pythagorean identity. Omitting that square changes the relationship, so the correct expression is $\\sin^2x=1-\\cos^2x$."}]},{"id":"66aff3db-0aaf-4ef2-83e9-bcf9f1c702a3","cardType":"cloze","points":null,"front":"The trigonometric identity used to convert $\\cos^2 x$ into terms of sine is $\\text{[...]}$.","back":"The trigonometric identity used to convert $\\cos^2 x$ into terms of sine is $\\boxed{\\cos^2 x = 1 - \\sin^2 x}$.","mcqOptions":[{"text":"$$\\cos^2 x = 1 - \\sin^2 x$","correct":true,"explanation":"This follows directly from the Pythagorean identity $\\sin^2 x + \\cos^2 x = 1$, rearranged to make $\\cos^2 x$ the subject."},{"text":"$$\\cos^2 x = 1 + \\sin^2 x$","correct":false,"explanation":"The sign should be negative when rearranging $\\sin^2 x + \\cos^2 x = 1$. Using a plus sign would make $\\cos^2 x$ too large and violate the identity."},{"text":"$$\\cos^2 x = \\sin^2 x - 1$","correct":false,"explanation":"Rearranging the Pythagorean identity requires subtracting $\\sin^2 x$ from $1$, not subtracting $1$ from $\\sin^2 x$. The correct result is $\\cos^2 x = 1 - \\sin^2 x$."},{"text":"$$\\cos^2 x = \\frac{1}{\\sin^2 x}$","correct":false,"explanation":"This confuses the Pythagorean identity with a reciprocal identity. The reciprocal of $\\sin^2 x$ is $\\cosec^2 x$, whereas the required conversion is $\\cos^2 x = 1 - \\sin^2 x$."}]},{"id":"d4fa36d1-c5d4-4c89-8e36-09738ae06305","cardType":"basic","points":2,"front":"Why is it dangerous to divide both sides of $2\\sin x = 3\\sin x\\cos x$ by $\\sin x$?","back":"It results in the loss of valid solutions where $\\sin x = 0$.","mcqOptions":[{"text":"It results in the loss of valid solutions where $\\sin x = 0$.","correct":true,"explanation":"Dividing by $\\sin x$ assumes it is non-zero, so any solutions with $\\sin x=0$ disappear. These values must be checked separately before dividing."},{"text":"It introduces extraneous solutions where $\\sin x = 0$.","correct":false,"explanation":"Division by $\\sin x$ cannot create solutions at zero; it removes them because division by zero is undefined. The valid solutions with $\\sin x=0$ are therefore lost."},{"text":"It makes the equation invalid because $\\cos x$ cannot equal zero.","correct":false,"explanation":"The issue is not whether cosine is zero. Dividing by $\\sin x$ excludes precisely the cases where $\\sin x=0$, so valid solutions must be retained separately."},{"text":"It changes the period of the equation by a factor of $2$.","correct":false,"explanation":"Dividing by a trigonometric factor does not automatically double the period. The real danger is excluding the valid zero-sine solutions, because division by zero is not permitted."}]},{"id":"144602a5-68dc-4a94-8ebd-e7edcffc6901","cardType":"basic","points":3,"front":"What is the preferred method to solve $2\\tan x = 3\\sin x$ without losing solutions?","back":"

Rewrite $\\tan x$ as $\\frac{\\sin x}{\\cos x}$
Multiply by $\\cos x$
Factorise the expression

","mcqOptions":[{"text":"

Rewrite $\\tan x$ as $\\frac{\\sin x}{\\cos x}$
Multiply by $\\cos x$
Factorise the expression

","correct":true,"explanation":"Replacing tangent gives $2\\frac{\\sin x}{\\cos x}=3\\sin x$. Multiplying by cosine and factorising preserves the possible zero factors, so solutions are not lost through division."},{"text":"

Rewrite $\\tan x$ as $\\frac{\\sin x}{\\cos x}$
Multiply by $\\sin x$
Expand the expression

","correct":false,"explanation":"Multiplying by sine and expanding does not expose the common factors needed to preserve every solution. Rewrite tangent, multiply by cosine, then factorise; this avoids dividing by a potentially zero factor."},{"text":"

Rewrite $\\tan x$ as $\\frac{\\cos x}{\\sin x}$
Multiply by $\\cos x$
Factorise the expression

","correct":false,"explanation":"Tangent is opposite over adjacent, so $\\tan x=\\frac{\\sin x}{\\cos x}$, not its reciprocal. Swapping the ratio changes the equation before factorisation and gives incorrect solutions."},{"text":"

Rewrite $\\tan x$ as $\\frac{\\sin x}{\\cos x}$
Divide by $\\cos x$
Factorise the expression

","correct":false,"explanation":"After replacing tangent, multiplying by cosine clears the denominator. Dividing by cosine risks excluding solutions where $\\cos x=0$, so it is not the safe method."}]},{"id":"2d0baa56-be89-4ba6-aa44-592496b7c463","cardType":"basic","points":1,"front":"If a quadratic trig equation yields $\\cos x = -2$, what action should be taken?","back":"Reject that solution because cosine cannot be less than $-1$.","mcqOptions":[{"text":"Reject that solution because cosine cannot be less than $-1$.","correct":true,"explanation":"For every real angle, cosine lies in the interval $-1\\leq\\cos x\\leq1$. Therefore $\\cos x=-2$ is impossible and must be rejected."},{"text":"Accept that solution because cosine can take any negative value.","correct":false,"explanation":"Cosine is bounded between $-1$ and $1$, so it cannot equal $-2$. The algebraic root must be rejected rather than converted into an angle."},{"text":"Reject that solution because cosine cannot be greater than $1$.","correct":false,"explanation":"The conclusion to reject is right, but the stated reason concerns values above $1$. Here the problem is that $-2$ is below the minimum possible cosine value of $-1$."},{"text":"Replace that solution with $\\cos x=-1$.","correct":false,"explanation":"An impossible quadratic root should not be rounded or replaced by the nearest boundary value. Since cosine cannot equal $-2$, the root is simply rejected."}]},{"id":"71615fed-5d75-452a-9089-37fe800958fe","cardType":"cloze","points":null,"front":"In the expression $\\tan^2 x$, the [ ] is squared, not the angle ($x$).","back":"In the expression $\\tan^2 x$, the ratio (tangent value) is squared, not the angle ($x$).","mcqOptions":[{"text":"ratio (tangent value)","correct":true,"explanation":"The superscript applies to the value of the tangent ratio: $\\tan^2x$ means $(\\tan x)^2$. It does not mean that the angle itself is squared."},{"text":"angle ($x$)","correct":false,"explanation":"This is the common mistake of treating the superscript as applying to the angle. In $\\tan^2x$, the tangent value is squared, while the angle remains $x$."},{"text":"angle ($2x$)","correct":false,"explanation":"A superscript does not double the angle. The notation means $(\\tan x)^2$, whereas doubling the angle would be written $\\tan(2x)$."},{"text":"period ($180^\\circ$)","correct":false,"explanation":"The superscript does not refer to tangent's period. It indicates that the tangent ratio is squared, while the period remains $180^\\circ$."}]},{"id":"9866e8a6-0392-4c54-8842-3701688613a8","cardType":"cloze","points":null,"front":"In the expression $\\tan^2 x$, the ratio (tangent value) is squared, not the [ ].","back":"In the expression $\\tan^2 x$, the ratio (tangent value) is squared, not the angle ($x$).","mcqOptions":[{"text":"angle ($x$)","correct":true,"explanation":"In $\\tan^2x$, the superscript applies to the tangent ratio, not to the input. Thus the angle remains $x$, and the expression means $(\\tan x)^2$."},{"text":"ratio ($\\tan x$)","correct":false,"explanation":"The ratio is already identified as the quantity being squared, so it cannot fill this blank. The contrasting quantity is the angle, which remains $x$."},{"text":"angle ($2x$)","correct":false,"explanation":"The notation does not double the input angle. $\\tan^2x$ means $(\\tan x)^2$, while an angle of twice the size would require $\\tan(2x)$."},{"text":"period ($180^\\circ$)","correct":false,"explanation":"The period is unrelated to the quantity named here. The superscript applies to the tangent value, and the unsquared input is the angle $x$."}]},{"id":"cd7e3c9b-6af0-47ca-8f2b-bd7949a67bb1","cardType":"basic","points":1,"front":"How do you find multiple solutions for a tangent equation once you have the principal value?","back":"Add or subtract multiples of $180^\\circ$ (the period).","mcqOptions":[{"text":"Add or subtract multiples of $180^\\circ$ (the period).","correct":true,"explanation":"Tangent repeats every $180^\\circ$, so adding or subtracting integer multiples of this period generates all angles with the same tangent value."},{"text":"Add or subtract multiples of $90^\\circ$ (the period).","correct":false,"explanation":"Although tangent has asymptotes separated by $180^\\circ$, its period is $180^\\circ$, not $90^\\circ$. Using $90^\\circ$ produces incorrect tangent values."},{"text":"Add or subtract multiples of $360^\\circ$ (the period).","correct":false,"explanation":"A $360^\\circ$ shift does return the same tangent, but it skips valid solutions between successive repeats. The fundamental tangent period is $180^\\circ$."},{"text":"Add or subtract multiples of $180^\\circ$ (the quadrant).","correct":false,"explanation":"The numerical interval is right, but $180^\\circ$ is tangent's period, not a quadrant. Calling it a quadrant confuses the geometric terminology and does not state the required reason."}]},{"id":"bc577015-0ce9-4804-92e2-afbcc537cc2f","cardType":"basic","points":2,"front":"When solving $\\sin(3\\theta-20^\\circ)=0.5$, at what stage should you add $20^\\circ$ and divide by $3$?","back":"Only after finding all possible values for the whole angle $(3\\theta-20^\\circ)$ in the transformed interval.","mcqOptions":[{"text":"Only after finding all possible values for the whole angle $(3\\theta-20^\\circ)$ in the transformed interval.","correct":true,"explanation":"First solve for every possible value of the complete angle $(3\\theta-20^\\circ)$. Then add $20^\\circ$ and divide by $3$ to obtain the corresponding values of $\\theta$."},{"text":"Only before finding all possible values for the whole angle $(3\\theta-20^\\circ)$ in the transformed interval.","correct":false,"explanation":"Applying the inverse linear operations too early can make the interval handling unclear and omit branches. Find all transformed-angle solutions first, then add $20^\\circ$ and divide by $3$."},{"text":"Before applying the inverse sine function to the value $0.5$.","correct":false,"explanation":"The inverse sine step is used first to find possible values of the complete angle. The operations involving $20^\\circ$ and division by $3$ come afterwards for each branch."},{"text":"Only after dividing the transformed interval by $3$, but before finding the second sine solution.","correct":false,"explanation":"Both sine branches must be found for the whole angle before transforming them. Dividing the interval before finding the second solution can cause a valid branch to be missed."}]},{"id":"afe777f1-bae6-4a9b-8bfe-97d3ac7ae654","cardType":"cloze","points":null,"front":"In an interval like $0 \\le x < 360^\\circ$, the value $0^\\circ$ is [ ] and $360^\\circ$ is excluded.","back":"In an interval like $0 \\le x < 360^\\circ$, the value $0^\\circ$ is included and $360^\\circ$ is excluded.","mcqOptions":[{"text":"included","correct":true,"explanation":"The lower bound uses ≤, so the endpoint at 0° belongs to the interval. The upper bound uses <, so 360° is not included."},{"text":"excluded","correct":false,"explanation":"The lower bound uses ≤ rather than <, so 0° belongs to the interval. The excluded endpoint is 360°, not 0°; this reverses the two boundary conditions."},{"text":"open","correct":false,"explanation":"An open endpoint means the value is excluded, but the interval includes its lower endpoint because it uses ≤. Only 360° is excluded here."},{"text":"greater","correct":false,"explanation":"The interval does not include values above its lower endpoint as a special rule; 0° is included exactly because the inequality is ≤, while 360° is excluded."}]},{"id":"225f8333-c594-4fad-97ac-9648b2e65638","cardType":"cloze","points":null,"front":"In an interval like $0 \\le x < 360^\\circ$, the value $0^\\circ$ is included and $360^\\circ$ is [ ].","back":"In an interval like $0 \\le x < 360^\\circ$, the value $0^\\circ$ is included and $360^\\circ$ is excluded.","mcqOptions":[{"text":"excluded","correct":true,"explanation":"The upper bound uses <, which means the endpoint 360° is not part of the interval. In contrast, 0° is included because the lower bound uses ≤."},{"text":"included","correct":false,"explanation":"The lower endpoint is included because its inequality is ≤, but the upper endpoint is excluded because the inequality is <. Swapping these gives the wrong interval."},{"text":"negative","correct":false,"explanation":"The endpoint is not omitted because it is negative; 360° is excluded specifically by the strict upper-bound inequality, while all values below it are allowed."},{"text":"closed","correct":false,"explanation":"A closed upper endpoint would require ≤, but the interval has < at 360°. Therefore the upper endpoint is excluded rather than closed."}]},{"id":"be9bcf0f-dc4b-494a-92ad-957d1a4cd548","cardType":"basic","points":2,"front":"Solve $3\\cos^2 x + 5\\cos x - 2 = 0$ given $(3\\cos x - 1)(\\cos x + 2) = 0$. Which factor provides valid solutions?","back":"$$3\\cos x - 1 = 0$ (or $\\cos x = \\frac{1}{3}$), because $\\cos x = -2$ is impossible.","mcqOptions":[{"text":"$$3\\cos x - 1 = 0$ (or $\\cos x = \\frac{1}{3}$), because $\\cos x = -2$ is impossible.","correct":true,"explanation":"The factor involving cosine can equal zero when $rac{1}{3}$ is a possible cosine value. The other factor would require $\\\\cos x=-2$, outside the range $[-1,1]$."},{"text":"$$\\cos x + 2 = 0$ (or $\\cos x = -2$), because $\\cos x = \\frac{1}{3}$ is impossible.","correct":false,"explanation":"This choice treats an impossible cosine value as valid. Every real cosine lies between -1 and 1, so the valid factor is $3\\cos x-1=0$, giving $\\cos x=\\frac{1}{3}$."},{"text":"Both factors provide valid solutions, because $\\cos x = \\frac{1}{3}$ and $\\cos x = -2$ are both possible.","correct":false,"explanation":"Both factors should not be accepted automatically after factorising. The cosine range rejects $\\cos x=-2$, leaving $3\\cos x-1=0$ and hence $\\cos x=\\frac{1}{3}$."},{"text":"Neither factor provides valid solutions, because the factorisation cannot be used to solve the trigonometric equation.","correct":false,"explanation":"The constant factor does not create a trigonometric solution, and the remaining factor must be checked against the cosine range. The valid condition is $3\\cos x-1=0$, so $\\cos x=\\frac{1}{3}$."}]},{"id":"5494c922-dd2a-4efb-a3d2-3de2be175602","cardType":"basic","points":2,"front":"How should you treat $2x$ when solving $2\\sin(2x)\\tan(2x)=\\cos(2x)+1$ for $0 \\le x < 180^\\circ$?","back":"Solve for the angle $u = 2x$ using the interval $0 \\le u < 360^\\circ$, then divide by $2$.","mcqOptions":[{"text":"Solve for the angle $u = 2x$ using the interval $0 \\le u < 360^\\circ$, then divide by $2$.","correct":true,"explanation":"Letting $u=2x$ converts the doubled angle into the variable being solved. As $0\\le x<180^\\circ$, doubling the bounds gives $0\\le u<360^\\circ$, after which each solution is halved."},{"text":"Solve for the angle $u = 2x$ using the interval $0 \\le u < 180^\\circ$, then divide by $2$.","correct":false,"explanation":"Using the original interval for $u$ would omit solutions because $u$ reaches almost $360^\\circ$ when $x$ reaches almost $180^\\circ$. Solve over $0\\le u<360^\\circ$, then divide by 2."},{"text":"Solve directly for $x$ using the interval $0 \\le x < 180^\\circ$, without changing the angle or interval.","correct":false,"explanation":"The doubled angle must first be solved over a doubled interval, not by leaving the original variable unchanged. Use $0\\le u<360^\\circ$, then divide every resulting angle by 2."},{"text":"Solve for the angle $u = 2x$ using the interval $0 \\\\le u < 360^\\circ$, then multiply by $2$.","correct":false,"explanation":"Doubling the interval is necessary when replacing $2x$ by $u$, but the final values of $x$ are obtained by halving the solved $u$ values, not doubling them again."}]},{"id":"d1c2cbb6-3eab-4bb7-b6df-04f6204f6b7a","cardType":"cloze","points":null,"front":"To solve equations mixing $\\sin^2 x$ and $\\cos x$, use the identity $\\text{[...]}$ to create a quadratic in $\\cos x$.","back":"To solve equations mixing $\\sin^2 x$ and $\\cos x$, use the identity $\\boxed{\\sin^2 x + \\cos^2 x = 1}$ to create a quadratic in $\\cos x$.","mcqOptions":[{"text":"$$\\sin^2 x + \\cos^2 x = 1$","correct":true,"explanation":"The Pythagorean identity replaces $\\sin^2 x$ with $1-\\cos^2 x$, allowing every term to be expressed in $\\cos x$. The equation can then be treated as a quadratic."},{"text":"$$\\sin^2 x - \\cos^2 x = 1$","correct":false,"explanation":"This identity is not the Pythagorean relationship for sine and cosine, so it cannot consistently eliminate $\\sin^2 x$. The required identity is $\\sin^2 x+\\cos^2 x=1$."},{"text":"$$\\sin^2 x + \\cos^2 x = 0$","correct":false,"explanation":"Replacing the sum by zero incorrectly suggests the squared ratios cancel. Their sum is one, which is the identity needed to rewrite $\\sin^2 x$ in terms of $\\cos x$."},{"text":"$$\\tan^2 x + 1 = \\sec^2 x$","correct":false,"explanation":"The reciprocal identity involving tangent does not relate these two squared functions in the required way. Use $\\sin^2 x+\\cos^2 x=1$ to form a quadratic in cosine."}]},{"id":"4b04687e-a42d-4f74-8fa8-65e3e72989dc","cardType":"cloze","points":null,"front":"To solve equations mixing $\\sin^2 x$ and $\\cos x$, use the identity $\\sin^2 x + \\cos^2 x = 1$ to create a quadratic in $\\text{[...]}$.","back":"To solve equations mixing $\\sin^2 x$ and $\\cos x$, use the identity $\\sin^2 x + \\cos^2 x = 1$ to create a quadratic in $\\boxed{\\cos x}$.","mcqOptions":[{"text":"$$\\cos x$","correct":true,"explanation":"Using $\\sin^2 x=1-\\cos^2 x$ leaves terms involving only cosine. The resulting equation can therefore be rearranged into a quadratic in $\\cos x$."},{"text":"$$\\sin x$","correct":false,"explanation":"The original equation already contains $\\cos x$, so the identity is used to eliminate $\\sin^2 x$, not to create a quadratic in sine. The target quantity is $\\cos x$."},{"text":"$$x$","correct":false,"explanation":"The identity can produce a quadratic in cosine because $\\sin^2 x$ becomes $1-\\cos^2 x$. It does not directly create a quadratic in the angle itself."},{"text":"$$\\tan x$","correct":false,"explanation":"Tangent is not the remaining variable after substituting the Pythagorean identity. The mixed sine-squared and cosine equation is converted into a quadratic in $\\cos x$."}]},{"id":"7d54ec15-c31b-4950-9b05-ce9a8e70af6f","cardType":"basic","points":1,"front":"What calculator mode is required for the angles discussed in this topic?","back":"Degree mode.","mcqOptions":[{"text":"Degree mode.","correct":true,"explanation":"The angles in this topic are measured in degrees, so the calculator must interpret trigonometric inputs such as 30 as 30°, not 30 radians."},{"text":"Radian mode.","correct":false,"explanation":"Radian mode interprets an input such as 30 as 30 radians, producing different trigonometric values. These questions use degree measures, so Degree mode is required."},{"text":"Gradian mode.","correct":false,"explanation":"Gradian mode uses a 400-unit turn rather than a 360-degree turn. Since the stated intervals and angles are in degrees, the calculator must use Degree mode."},{"text":"Automatic mode.","correct":false,"explanation":"Automatic mode is not a standard angle unit that guarantees degree interpretation. Select Degree mode explicitly so the calculator matches the degree intervals used in these equations."}]},{"id":"db2202f7-306d-4806-8985-fcb482b87708","cardType":"basic","points":1,"front":"Why can a single trig equation have multiple solutions within an interval?","back":"Because trigonometric graphs are periodic (they repeat).","mcqOptions":[{"text":"Because trigonometric graphs are periodic (they repeat).","correct":true,"explanation":"Sine, cosine and tangent repeat their values after fixed periods, so a horizontal interval can contain several angles producing the same result. Each valid angle must therefore be found."},{"text":"Because trigonometric graphs are linear (they have constant gradients).","correct":false,"explanation":"A trig function is not one-to-one over a full interval because its graph repeats. Multiple solutions arise from periodicity, not because the equation has several unrelated functions."},{"text":"Because trigonometric graphs are quadratic (they have two roots).","correct":false,"explanation":"The presence of a squared term can create multiple algebraic roots, but even unsquared trigonometric equations can have several solutions because the graphs repeat periodically."},{"text":"Because trigonometric graphs are random (they take repeated values unpredictably).","correct":false,"explanation":"The interval does not itself generate repeated solutions. Multiple angles occur because trigonometric functions repeat their values, which is described as periodic behaviour."}]},{"id":"468b61f4-1151-40bd-8907-b6e38b549efb","cardType":"basic","points":2,"front":"What are the two components used to find solutions beyond the principal value?","back":"Symmetry and periodicity of the trig graph.","mcqOptions":[{"text":"Symmetry and periodicity of the trig graph.","correct":true,"explanation":"The trig graph’s symmetry provides related angles with the same or linked values, while periodicity allows the pattern to repeat beyond one cycle. Together, they generate solutions beyond the principal value."},{"text":"Amplitude and phase shift of the trig graph.","correct":false,"explanation":"Amplitude and phase shift describe transformations of a trig graph, but they do not generate further angle solutions. The required components are symmetry and periodicity of the trig graph."},{"text":"Gradient and intercept of the trig graph.","correct":false,"explanation":"The gradient and intercept are key features of linear graphs, not the features used to find repeated trigonometric solutions. The correct components are symmetry and periodicity of the trig graph."},{"text":"Domain and range of the trig graph.","correct":false,"explanation":"Domain and range restrict possible inputs and outputs, but they do not identify equivalent angles or repeated cycles. The required components are symmetry and periodicity of the trig graph."}]}]}],"examCountdown":{"examDate":"2027-05-13","paperLabel":"Paper 1"},"heading":"10.4 Solving Trigonometric Equations","title":"10.4 Solving Trigonometric Equations Flashcards","topbar":{"backHref":"/as-level/maths/edexcel/flashcards"},"topicSidebar":"$L39"}]}],"topicId":"218c4c01-fee8-4bde-b4a1-16e52b127ead","walled":"$L3a"}],"$L3b","$L3c"]