Understanding Engineering Mathematics

6.1.7 Compound angle formulae ➤187 196➤➤

A.Expand sin(A+B)in terms of sine and cosine ofAandB.
B. FromAderive similar expansions for

(i) sin(A−B) (ii) cos(A−B) (iii) tan(A+B)

C.Given that sin 45°=cos 45°=

1

√

2

, cos 60°=

1

2

,sin60°=

√

3

2

,evaluate

(i) cos 75° (ii) sin 105° (iii) tan(− 75 °)

D.Express cos 2Ain trigonometric ratios ofA.

E. Given that cos 30°=

√

3

2

,sin30°=

1

2

,evaluate

(i) sin 15° (ii) tan 15°

6.1.8 Trigonometric equations ➤191 196➤➤

Find the general solution of each of the equations:

(i) sinθ+2sinθcosθ= 0

(ii) cos 3θ=cosθ

6.1.9 TheacosqYbsinqform ➤192 197➤➤

Express cosθ+sinθin the form (i)rsin(θ+α) (ii)rcos(θ+α)

6.2 Revision

6.2.1 Radian measure and the circle

➤

171 194➤

As noted in Section 5.2.2, aradianis the angle subtended at the centre of a circle by an
arc with length equal to that of the radius. It follows that

θradians =

180

π

θdegrees

Thelengthofarcofacircleofradiusr, subtending angleθ radians at the centre is
by definitions=rθ. See Figure 6.1. The area of the enclosed sector isA=^12 r^2 θ.This
follows because the total area of the circle isπr^2 and a sector with angleθradians forms

a fraction

θ

2 π

of that area.