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Compound Angle Equations - Questions

Question 1
26214

In the domain \(0 \le \theta \le 2\pi \), find the solutions to \(\cos 2\theta - \sin \theta = 0\). 

\(\theta = \dfrac{\pi }{6},\,\dfrac{{5\pi }}{6}\) or \(\dfrac{{3\pi }}{2}\)

Question 2
16090

For \(0 \le \theta \le 2\pi \), solve \(\tan 2\theta = 3\tan \theta \)

\(\theta = 0,\dfrac{\pi }{6},\dfrac{{5\pi }}{6},\pi ,\dfrac{{7\pi }}{6},\dfrac{{11\pi }}{6},2\pi \)

Question 3
16091

For \(0^\circ \le \theta \le 2\pi \), solve \({\sin ^2}2\theta = {\cos ^2}\theta \)

\(\theta = \dfrac{\pi }{6},\dfrac{{5\pi }}{6},\dfrac{\pi }{2},\dfrac{{7\pi }}{6},\dfrac{{3\pi }}{2},\dfrac{{11\pi }}{6}\)

Question 4
30484

Show that \(\displaystyle \,\text{cosec}\, 2\theta +\cot 2\theta = \cot \theta\). Use this result:

a) To find in surd form \(\displaystyle \cot \frac{\pi}{8} \text{ and } \cot \frac{\pi}{12}\)

b) To show, without using a calculator, that:

\[\,\text{cosec}\, \frac{4\pi}{15} + \,\text{cosec}\,\frac{8\pi}{15} + \,\text{cosec}\, \frac{16\pi}{15} + \,\text{cosec}\, \frac{32\pi}{15} = 0\]

a) \(\cot \dfrac{\pi}{8} = 1+\sqrt{2},\quad \cot\dfrac{\pi}{12} = 2+\sqrt{3}\)   b) True

Question 5
30616

i) Prove that \(\tan 4\theta - \tan\theta = \tan 3\theta (1+\tan 4\theta\tan \theta)\)

ii) Using part (i), or otherwise, prove that

\[\tan 4\theta - \tan\theta = \frac{\sin 3\theta}{\cos 4\theta \cos \theta}\]

iii) Hence solve for \(0 \leq \theta \leq \pi\).

\[\tan 4\theta = \tan\theta\]

i) True ii) True iii) \(\theta = 0,\, \dfrac{\pi}{3},\,\dfrac{2\pi}{3},\,\pi\)

Question 6
30608

Prove that

i) \(\cot\theta +\tan\theta = 2\, \text{cosec}\, 2\theta\)

ii) \(\cot\theta - \tan\theta =2\cot 2\theta\)

iii) Hence find, in exact form the values of \(\cot\dfrac{\pi}{12}\) and \(\tan\dfrac{\pi}{12}\).

i) True  ii) True  iii) \(\cot\dfrac{\pi}{12}=2+\sqrt3\) and \(\tan\dfrac{\pi}{12}=2-\sqrt3\)

Question 7
30603

Given that \(\sin\theta + \cos\theta \neq 0\),

i) Prove that \(\dfrac{2\sin^3 \theta + 2\cos^3\theta}{\sin\theta + \cos\theta} = 2 - \sin (2\theta)\)

ii) Hence or otherwise solve \(\dfrac{2\sin^3 \theta + 2\cos^3\theta}{\sin\theta + \cos\theta} = 1\) in the domain \(-\pi \leq \theta \leq \pi\).

i) True ii) True

Question 8
12476

In the domain \(0^\circ \le x \le 360^\circ \), find the solutions to \(\sin 2x - \cos x = 0\)

\(x = 30^\circ ,\,\,90^\circ ,\,\,150^\circ ,\,\,270^\circ \)

Question 9
12477

In the domain \( - 180^\circ \le \theta \le 180^\circ \), find the solutions to \(\cos 2\theta - \cos \theta = 0\)

\(\theta = - 120^\circ ,\,\,0^\circ ,\,\,120^\circ \)

Question 10
12478

In the domain \(0 \le x \le 2\pi \), find the solutions to \(tan2x + \tan x = 0\)

\(x = 0,\,\,\dfrac{\pi }{3},\,\,\dfrac{{2\pi }}{3},\,\,\pi ,\,\,\dfrac{{4\pi }}{3},\,\,\dfrac{{5\pi }}{3},\,\,2\pi \)

Question 11
12479

In the domain, \( - \pi \le \theta \le \pi \), find the solutions to \(\cos 2\theta - \cos \theta + 1 = 0\)

\(\theta = \dfrac{{ - \pi }}{2},\,\,\dfrac{{ - \pi }}{3},\,\,\dfrac{\pi }{3},\,\,\dfrac{\pi }{2}\)

Question 12
12480

In the domain, \(0 \le x \le \pi \), find the solutions to \(\sin 2x + \sin x = \sin 3x\)

\(x = 0,\,\,\dfrac{{2\pi }}{3},\,\,\pi \)

Question 13
52021

In the domain \(-\pi \le \theta \le \pi\), find the solutions to \(2 \cos 2 \theta = 1-4 \cos \theta\)

\(\theta=\dfrac{\pi}{3} \text { or }-\dfrac{\pi}{3}\)

Question 14
64529

(i) Use the expansion for \(\sin (A+B)\) and the exact values for \(\cos \dfrac{\pi}{4}\) and \(\sin \dfrac{\pi}{4}\) to show that \(\sin \left(x+\dfrac{\pi}{4}\right)=\dfrac{\sin x+\cos x}{\sqrt{2}}\).

(ii) Hence, or otherwise, solve \(\dfrac{\sin x+\cos x}{\sqrt{2}}=\dfrac{\sqrt{3}}{2}\) for \(0 \leq x \leq 2 \pi\).

i) True

ii) \(x=\dfrac{\pi}{12} \text { or } \dfrac{5 \pi}{12} \quad(0 \leq x \leq 2 \pi)\)