3=0$"},{"latex":"$$\\color{@purple}{4\\cos^2 x}+\\cos x-3=0$"}]},"order":3,"title":"Collect the terms","description":"Expand the right-hand side, move every term to the left, and collect like terms. This gives the required form and earns A1."}]}],"generatedAt":"2026-07-20T16:30:00.176Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"9e7bf032-79e0-49db-b07f-29e5855afe38","label":"b","content":"Hence solve, for $0 \\leq x < 360^\\circ$, the equation,\n\n$$1 + \\cos x = 3 \\tan x \\sin x$$\n\nGive your answers to one decimal place where appropriate.","markscheme":"- Factorises: $(4\\cos x - 3)(\\cos x + 1) = 0$ M1\n- $\\cos x = 3/4$ or $\\cos x = -1$ M1\n- For $\\cos x = 3/4$, $x = 41.4^\\circ, 318.6^\\circ$ A2\n- For $\\cos x = -1$, $x = 180^\\circ$ A1","marks":5,"order":1,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Attempts to factorise the quadratic.","code":"M"},{"id":"p2","marks":1,"description":"Finds $\\cos x = 0.75$ and $\\cos x = -1$.","code":"M"},{"id":"p3","marks":1,"description":"Finds $x = 41.4^\\circ$.","code":"A"},{"id":"p4","marks":1,"description":"Finds $x = 318.6^\\circ$.","code":"A"},{"id":"p5","marks":1,"description":"Finds $x = 180^\\circ$.","code":"A"}],"finalAnswers":["41.4","180","318.6"],"generatedAt":"2026-05-12T02:14:06.607Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"trig_factorisation","label":"Factorise the trigonometric equation","steps":[{"type":"text","order":0,"title":"Check the domain","content":"Since $\\tan x=\\dfrac{\\sin x}{\\cos x}$, the original equation is defined only when $\\cos x\\ne 0$. Therefore $x=90^\\circ$ and $x=270^\\circ$ are excluded, and multiplying by $\\cos x$ is valid for every admissible solution. This gives\n\n$$\n(1+\\cos x)\\cos x=3\\sin^2x.\n$$","highlights":[{"text":"$$0 \\leq x < 360^\\circ$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\left(1+\\cos x\\right)\\cos x=3\\sin^2x$"},{"latex":"$$\\left(1+\\cos x\\right)\\cos x=3\\left(1-\\cos^2x\\right)$"},{"latex":"$$\\htmlData{anchor=quadratic}{\\color{@sky}{4\\cos^2x+\\cos x-3=0}}$"},{"latex":"$$\\htmlData{anchor=factorised}{\\color{@sky}{(4\\cos x-3)(\\cos x+1)=0}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"factorised"}],"connections":[{"to":"factorised","from":"quadratic","color":"sky","label":"factorise"}]},"order":1,"title":"Factorise the equation","description":"Use the identity ‘sin² x = 1 − cos² x’, collect all terms on one side, and factorise. The final factorisation earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=firstfactor}{\\color{@sky}{4\\cos x-3=0}}$"},{"latex":"$$\\htmlData{anchor=firstvalue}{\\color{@sky}{\\cos x=\\frac34}}$"},{"latex":"$$\\htmlData{anchor=secondfactor}{\\color{@orange}{\\cos x+1=0}}$"},{"latex":"$$\\htmlData{anchor=secondvalue}{\\color{@orange}{\\cos x=-1}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"firstvalue"},{"color":"orange","style":"text-color-with-background","anchor":"secondvalue"}],"connections":[{"to":"firstvalue","from":"firstfactor","color":"sky","label":"rearrange"},{"to":"secondvalue","from":"secondfactor","color":"orange","label":"rearrange"}]},"order":3,"title":"State the cosine values","description":"Set each factor equal to zero. Obtaining both cosine values earns M1."},{"type":"text","order":4,"title":"Solve the positive-cosine solutions","content":"For $\\cos x=\\frac34$, the reference angle is $\\cos^{-1}\\left(\\frac34\\right)=41.4^\\circ$. Cosine is positive in quadrants I and IV, so the solutions in the given interval are $x=41.4^\\circ$ and $x=318.6^\\circ$. This earns A2."},{"type":"text","order":5,"title":"Solve the second case","content":"For $\\cos x=-1$, the solution in $0\\leq x<360^\\circ$ is $x=180^\\circ$. This earns A1. Combining both cases, the solutions in ascending order are $$\\boxed{x=41.4^\\circ,\\ 180^\\circ,\\ 318.6^\\circ}$$. The e

Trigonometric Identities and Equations

Question 11

Trigonometric Identities and Equations Questions

Trigonometric Identities and Equations

Question 11

Trigonometric Identities and Equations Questions