Higher Engineering Mathematics

John Bird

Chapter 19

Compound angles - all with Video Answers

Educators

Chapter Questions

Problem 1

Expand and simplify the following expressions:
(a) $\sin (\pi+\alpha)$
(b) $-\cos \left(90^{\circ}+\beta\right)$
(c) $\sin (A-B)-\sin (A+B)$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 2

Prove that
$$
\cos (y-\pi)+\sin \left(y+\frac{\pi}{2}\right)=0
$$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 3

Show that
$$
\tan \left(x+\frac{\pi}{4}\right) \tan \left(x-\frac{\pi}{4}\right)=-1
$$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 4

If $\sin P=0.8142$ and $\cos Q=0.4432$ evaluate, correct to 3 decimal places:
(a) $\sin (P-Q)$ (b) $\cos (P+Q)$
(c) $\tan (P+Q)$, using the compound angle formulae.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 5

Solve the equation
$$
4 \sin \left(x-20^{\circ}\right)=5 \cos x
$$
for values of $x$ between $0^{\circ}$ and $90^{\circ}$

Steven Clarke

Numerade Educator

Problem 6

Find an expression for $3 \sin \omega t+4 \cos \omega t$ in the form $R \sin (\omega t+\alpha)$ and sketch graphs of $3 \sin \omega t, 4 \cos \omega t$ and $R \sin (\omega t+\alpha)$ on the same axes.

AG

Ankit Gupta

Numerade Educator

Problem 7

Express $4.6 \sin \omega t-7.3 \cos \omega t$ in the form $R \sin (\omega t+\alpha)$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 8

Express $-2.7 \sin \omega t-4.1 \cos \omega t$ in the form $R \sin (\omega t+\alpha)$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 9

Express $3 \sin \theta+5 \cos \theta$ in the form $R \sin (\theta+\alpha)$, and hence solve the equation $3 \sin \theta+5 \cos \theta=4$, for values of $\theta$ between $0^{\circ}$ and $360^{\circ}$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 10

Solve the equation $3.5 \cos A-5.8 \sin A=6.5$ for $0^{\circ} \leq A \leq 360^{\circ}$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 11

$I_{3} \sin 3 \theta$ is the third harmonic of a waveform. Express the third harmonic in terms of the first harmonic $\sin \theta$, when $I_{3}=1$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 12

Prove that $\frac{1-\cos 2 \theta}{\sin 2 \theta}=\tan \theta$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 13

Prove that
$$
\cot 2 x+\operatorname{cosec} 2 x=\cot x
$$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 14

Solve the equation $\cos 2 \theta+3 \sin \theta=2$ for $\theta$ in the range $0^{\circ} \leq \theta \leq 360^{\circ}$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 15

Express $\sin 4 x \cos 3 x$ as a sum or difference of sines and cosines.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 16

Express $2 \cos 5 \theta \sin 2 \theta$ as a sum or difference of sines or cosines.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 17

Express $3 \cos 4 t \cos t$ as a sum or difference of sines or cosines.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 18

In an alternating current circuit, voltage $v=5 \sin \omega t$ and current $i=10 \sin (\omega t-$ $\pi / 6$ ). Find an expression for the instantaneous power $p$ at time $t$ given that $p=v i$, expressing the answer as a sum or difference of sines and cosines.

Khushbu Rani

Numerade Educator

Problem 19

Express $\sin 5 \theta+\sin 3 \theta$ as a product.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 20

Express $\sin 7 x-\sin x$ as a product.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 21

Express $\cos 2 t-\cos 5 t$ as a product.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 22

Show that $\frac{\cos 6 x+\cos 2 x}{\sin 6 x+\sin 2 x}=\cot 4 x$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 23

Solve the equation $\cos 4 \theta+\cos 2 \theta=0$ for $\theta$ in the range $0^{\circ} \leq \theta \leq 360^{\circ}$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator