atio}{\\color{@sky}{\\frac{3}{5}}}"},{"latex":"\\alpha=\\htmlData{anchor=inverse}{\\color{@sky}{\\tan^{-1}\\!\\left(\\frac{3}{5}\\right)}}"},{"latex":"\\alpha\\approx31.0^\\circ"}],"connections":[{"to":"inverse","from":"ratio","color":"sky","label":"inverse tangent","style":"solid"}]},"order":1,"title":"Find the reference angle","description":"Use the inverse tangent to find the acute reference angle associated with the positive ratio. The quadrant check comes next."},{"type":"text","order":2,"title":"Choose the quadrants","content":"Since $\\tan x$ is positive, $x$ must lie in quadrants I or III. The first-quadrant solution is $x=31.0^\\circ$, earning the first A1; add $180^\\circ$ next to find the quadrant III solution.","highlights":[{"text":"$$0^\\circ \\leq x \\leq 360^\\circ$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=first}{x=31.0^\\circ}"},{"latex":"x=31.0^\\circ+\\htmlData{anchor=addend}{180^\\circ}"},{"latex":"x=\\htmlData{anchor=result}{211.0^\\circ}"},{"text":"Therefore, 31.0 degrees and 211.0 degrees are the complete solutions in the stated interval."}],"connections":[{"to":"addend","from":"first","color":"orange","label":"add 180 degrees","style":"solid"},{"to":"result","from":"addend","color":"sky","label":"simplify","style":"solid"}]},"order":3,"title":"Find the third-quadrant solution","description":"Add 180 degrees to the first-quadrant solution. This gives the second A1."}]}],"generatedAt":"2026-07-21T04:06:05.213Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"317b3927-b361-49e3-9a67-ec39dae616c3","revisionId":"696d20e6-77d0-4323-8e2d-a4994cd5c16a","questionNumber":"30","questionType":"LA","specification":"Solve the equation $24 \\tan x + 5 \\cos x = 0$ for $0^\\circ \\leq x \\leq 360^\\circ$, giving your answers to the nearest degree","difficulty":3,"difficultyBand":"medium","totalMarks":6,"sourceTitle":"Question Set 10: Trigonometric Identities and Equations","calculatorUse":null,"parts":[{"id":"721bae76-07ef-4873-95c3-ef00f5c4ee17","label":"0","content":"","markscheme":"- $24\\frac{\\sin x}{\\cos x} + 5\\cos x = 0 \\Rightarrow 24\\sin x + 5\\cos^2 x = 0$ M1\n- $24\\sin x + 5(1 - \\sin^2 x) = 0 \\Rightarrow 5\\sin^2 x - 24\\sin x - 5 = 0$ M1\n- $(5\\sin x + 1)(\\sin x - 5) = 0$ M1\n- $\\sin x = -1/5$ (reject $\\sin x = 5$) A1\n- $x = 180 - (-11.5) = 192^\\circ$ A1\n- $x = 360 - 11.5 = 348^\\circ$ A1","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Substitute $\\sin x / \\cos x$ and multiplies by $\\cos x$.","code":"M"},{"id":"p2","marks":1,"description":"Forms correct quadratic in $\\sin x$.","code":"M"},{"id":"p3","marks":1,"description":"Factorises and finds $\\sin x = -0.2$.","code":"M"},{"id":"p4","marks":1,"description":"Correctly rejects $\\sin x = 5$.","code":"B"},{"id":"p5","marks":1,"description":"Finds $x = 192^\\circ$.","code":"A"},{"id":"p6","marks":1,"description":"Finds $x = 348^\\circ$.","code":"A"}],"finalAnswers":["192","348"],"generatedAt":"2026-05-12T02:14:06.607Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"trig_identity","label":"Use trigonometric identities","steps":[{"type":"text","order":0,"title":"Rewrite tangent","content":"Use $\\tan x=\\dfrac{\\sin x}{\\cos x}$. Since the original equation contains $\\tan x$, values where $\\cos x=0$ are not in its domain; the relevant excluded angles in this interval are $90^\\circ$ and $270^\\circ$. Multiply through by $\\cos x$ to remove the denominator. This earns the first method mark.","highlights":[{"text":"$$24 \\tan x + 5 \\cos x = 0$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$24\\htmlData{anchor=tanratio}{\\dfrac{\\sin x}{\\cos x}}+5\\cos x=0$"},{"latex":"$$\\htmlData{anchor=cleared}{24\\sin x+5\\cos^2x=0}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"tanr

Trigonometric Identities and Equations

Question 24

Trigonometric Identities and Equations Questions

Trigonometric Identities and Equations

Question 24

Trigonometric Identities and Equations Questions