Questions on Trigonometric Functions of Multiple Angles

This page contains multiple-choice questions on evaluating trigonometric functions of multiple angles. Key formulas are provided first, followed by worked solutions for each question.

Multiple Angle Trigonometric Formulas

Common identities used to evaluate trigonometric functions of multiple angles:

\[ \sin(2x) = 2 \sin x \cos x \] \[ \sin(3x) = 3\sin x - 4\sin^3 x = \sin x(4\cos^2 x - 1) \] \[ \sin(4x) = 4\sin x \cos x(1 - 2\sin^2 x) \] \[ \cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 \] \[ \cos(3x) = 4\cos^3 x - 3\cos x \] \[ \cos(4x) = 8\cos^4 x - 8\cos^2 x + 1 \] \[ \tan(2x) = \frac{2\tan x}{1 - \tan^2 x} \] \[ \tan(3x) = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x} \]

Multiple Choice Questions

Question 1

If \( \sin t = 0.6 \) and \( \cot t > 0 \), find \( \sin(2t) \).

a) −0.96
b) 0.48
c) 0.96
d) −0.48

Question 2

If \( \cos t = 0.8 \), find \( \cos(2t) \).

a) 0.28
b) 0.4
c) 1.0
d) 1.6

Question 3

If \( \tan x = 5 \), find \( \tan(2x) \).

a) 10
b) −5/12
c) 1/10
d) 5/12

Question 4

If \( \cos t = \frac{3}{4} \) and \( \sin t < 0 \), find \( \sin(3t) \).

a) \( \frac{\sqrt{7}}{16} \)
b) \( -\frac{5\sqrt{7}}{16} \)
c) \( -\frac{3\sqrt{7}}{4} \)
d) \( \frac{5\sqrt{7}}{16} \)

Question 5

If \( \cos t = \frac{1}{3} \) and \( \frac{3\pi}{2} < t < 2\pi \), find \( \sin(4t) \).

a) \( \frac{8\sqrt{2}}{3} \)
b) \( -\frac{8\sqrt{2}}{3} \)
c) \( -\frac{56\sqrt{2}}{243} \)
d) \( \frac{56\sqrt{2}}{81} \)

Question 6

If \( \sin t = \frac{1}{5} \) and \( 0 < t < \frac{\pi}{2} \), find \( \cos(4t) \).

a) 0.3464
b) 0.8
c) 0.6928
d) −0.6928

Answers with Explanations

Answer: c)
Using \( \sin(2t) = 2\sin t \cos t \). Since \( \sin t = 0.6 \) and \( \cot t > 0 \), cosine is positive. \[ \cos t = \sqrt{1 - \sin^2 t} = \sqrt{1 - 0.36} = 0.8 \] \[ \sin(2t) = 2(0.6)(0.8) = 0.96 \]
Answer: a)
\[ \cos(2t) = 2\cos^2 t - 1 = 2(0.8)^2 - 1 = 0.28 \]
Answer: b)
\[ \tan(2x) = \frac{2\tan x}{1 - \tan^2 x} = \frac{10}{1 - 25} = -\frac{5}{12} \]
Answer: b)
\[ \sin t = -\sqrt{1 - \left(\frac{3}{4}\right)^2} = -\frac{\sqrt{7}}{4} \] \[ \sin(3t) = 3\sin t - 4\sin^3 t = -\frac{5\sqrt{7}}{16} \]
Answer: d)
Using quadrant IV signs and multiple-angle identities, simplifying gives: \[ \sin(4t) = \frac{56\sqrt{2}}{81} \]
Answer: c)
\[ \cos(4t) = 8\cos^4 t - 8\cos^2 t + 1 \] Using \( \sin t = \frac{1}{5} \Rightarrow \cos t = \frac{2\sqrt{6}}{5} \), substitution yields: \[ \cos(4t) \approx 0.6928 \]