Samuel has to solve the equation
8−3sinx=2+6cos2x 8 - 3\sin x = 2 + 6\cos^2 x 8−3sinx=2+6cos2xwhere −90∘≤x<270∘-90^{\circ} \le x < 270^{\circ}−90∘≤x<270∘
Samuel’s working is as follows:
8−3sinx=2+6cos2x6−3sinx=6cos2x6−3sinx=6(1−sin2x)6−3sinx=6−6sin2x−3sinx=−6sin2x3=6sinxsinx=0.5x=30∘ \begin{aligned} 8 - 3\sin x &= 2 + 6\cos^2 x \\ 6 - 3\sin x &= 6\cos^2 x \\ 6 - 3\sin x &= 6(1 - \sin^2 x) \\ 6 - 3\sin x &= 6 - 6\sin^2 x \\ -3\sin x &= -6\sin^2 x \\ 3 &= 6\sin x \\ \sin x &= 0.5 \\ x &= 30^{\circ} \end{aligned} 8−3sinx6−3sinx6−3sinx6−3sinx−3sinx3sinxx=2+6cos2x=6cos2x=6(1−sin2x)=6−6sin2x=−6sin2x=6sinx=0.5=30∘Explain the two errors that Samuel has made.
Write down all the values of x x\,x that satisfy the equation
8−3sinx=2+6cos2x 8 - 3\sin x = 2 + 6\cos^2 x 8−3sinx=2+6cos2xwhere −90∘≤x<270∘-90^{\circ} \le x < 270^{\circ}−90∘≤x<270∘
322 exam-style questions on Edexcel A Level Maths Trigonometric Identities and Equations, covering 10.1 Angles in all four Quadrants, 10.2 Exact Values of Trigonometric Ratios, 10.3 Trigonometric Identities, 10.4 Solving Trigonometric Equations, 10.5 Harder Trigonometric Equations, and 10.6 Equations and Identities. Each one has a worked solution and a mark scheme showing where the marks go.