Question
(a) Show that sin 2x + cos 2x – 1 = 2 sin x (cos x – sin x) . [2]
(b) Hence or otherwise, solve sin 2x + cos 2x – 1 + cos x – sin x = 0 for 0 < x < 2π . [6]
Answer/Explanation
Ans
Question
Solve \(\cos 2x – 3\cos x – 3 – {\cos ^2}x = {\sin ^2}x\) , for \(0 \le x \le 2\pi \) .
Answer/Explanation
Markscheme
evidence of substituting for \(\cos 2x\) (M1)
evidence of substituting into \({\sin ^2}x + {\cos ^2}x = 1\) (M1)
correct equation in terms of \(\cos x\) (seen anywhere) A1
e.g. \(2{\cos ^2}x – 1 – 3\cos x – 3 = 1\) , \(2{\cos ^2}x – 3\cos x – 5 = 0\)
evidence of appropriate approach to solve (M1)
e.g. factorizing, quadratic formula
appropriate working A1
e.g. \((2\cos x – 5)(\cos x + 1) = 0\) , \((2x – 5)(x + 1)\) , \(\cos x = \frac{{3 \pm \sqrt {49}}}{4}\)
correct solutions to the equation
e.g. \(\cos x = \frac{5}{2}\) , \(\cos x = – 1\) , \(x = \frac{5}{2}\) , \(x = – 1\) (A1)
\(x = \pi \) A1 N4
[7 marks]
Question
Show that \(4 – \cos 2\theta + 5\sin \theta = 2{\sin ^2}\theta + 5\sin \theta + 3\) .
Hence, solve the equation \(4 – \cos 2\theta + 5\sin \theta = 0\) for \(0 \le \theta \le 2\pi \) .
Answer/Explanation
Markscheme
attempt to substitute \(1 – 2{\sin ^2}\theta \) for \(\cos 2\theta \) (M1)
correct substitution A1
e.g. \(4 – (1 – 2{\sin ^2}\theta ) + 5\sin\theta \)
\(4 – \cos 2\theta + 5\sin \theta = 2{\sin ^2}\theta + 5\sin\theta + 3\) AG N0
[2 marks]
evidence of appropriate approach to solve (M1)
e.g. factorizing, quadratic formula
correct working A1
e.g. \((2\sin \theta + 3)(\sin \theta + 1)\) , \((2x + 3)(x + 1) = 0\) , \(\sin x = \frac{{ – 5 \pm \sqrt 1 }}{4}\)
correct solution \(\sin \theta = – 1\) (do not penalise for including \(\sin \theta = – \frac{3}{2}\) (A1)
\(\theta = \frac{{3\pi }}{2}\) A2 N3
[5 marks]