Jacob has to solve the equation
3−sinx=1+2cos2x 3 - \sin x = 1 + 2\cos^2 x 3−sinx=1+2cos2xwhere −180∘≤x<180∘-180^{\circ} \leq x < 180^{\circ}−180∘≤x<180∘
Jacob's working is as follows:
3−sinx=1+2cos2x2−sinx=2cos2x2−sinx=2(1−sin2x)2−sinx=2−2sin2x−sinx=−2sin2x1=2sinxsinx=0.5x=30∘ \begin{aligned} 3 - \sin x &= 1 + 2\cos^2 x \\ 2 - \sin x &= 2\cos^2 x \\ 2 - \sin x &= 2(1 - \sin^2 x) \\ 2 - \sin x &= 2 - 2\sin^2 x \\ -\sin x &= -2\sin^2 x \\ 1 &= 2\sin x \\ \sin x &= 0.5 \\ x &= 30^{\circ} \end{aligned} 3−sinx2−sinx2−sinx2−sinx−sinx1sinxx=1+2cos2x=2cos2x=2(1−sin2x)=2−2sin2x=−2sin2x=2sinx=0.5=30∘Explain the two errors that Jacob has made.
Write down all the values of x x\,x that satisfy the equation
3−sinx=1+2cos2x 3 - \sin x = 1 + 2\cos^2 x 3−sinx=1+2cos2xwhere −180∘≤x<180∘-180^{\circ} \leq x < 180^{\circ}−180∘≤x<180∘
Practise AQA A Level Maths 1.8 E: Trigonometry with exam-style questions for A Level Maths. 239 questions covering 1.8.1 Trigonometric definitions, rules and radians, 1.8.2 Small angle approximations (A-level only), 1.8.3 Trigonometric functions and exact values, 1.8.4 Reciprocal and inverse trigonometric functions (A-level only), 1.8.5 Trigonometric identities, 1.8.6 Compound and double angle formulae (A-level only), 1.8.7 Solving trigonometric equations, 1.8.8 Proofs with trigonometric identities, and 1.8.9 Trigonometry in context, matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
