Understanding Engineering Mathematics

(iv) sinθ=0(v)sinθ=−1(vi)sinθ=

√

3

(vii) tanθ= 0 (viii) tanθ=−1(ix)tanθ=

√

3

B.Find the general solutions of the equations

(i) cos 2θ= 1 (ii) sin 2θ=sinθ

(iii) cos 2θ+sinθ=0(iv)cos2θ+cos 3θ= 0

(v) sec^2 θ=3tanθ− 1

6.3.9 TheacosqYbsinqform

➤➤

173 192

➤

A.Write the following in the form (a)rsin(θ+α),(b)rcos(θ+α)

(i) sinθ−cosθ (ii)

√

3cosθ+sinθ (iii)

√

3sinθ−cosθ

(iv) 3 cosθ+4sinθ

B. Determine the maximum and minimum values of each of the expressions in Ques-
tionA, stating the values where they occur, in the range 0≤θ≤ 2 π.
C.Find the solutions, in the range 0≤θ≤ 2 π, of the equations obtained by equating the
expressions in QuestionAto (a) 1 (b)−1.

6.4 Applications

1.In the theory of the elasticity of solids a crucial step in establishing the connection
between the elastic constants and Poisson’s ratio is to determine a relation between
the normal strain in the 45°direction,ε, and the shear strain for a square section of
the sort shown in Figure 6.14. By considering Figure 6.14b, where the diagonalBD
is stretched by a factor 1+ε, and using the fact that for small anglesγ(in radians)
sinγ∼=γ, show that

ε∼=

γ

2

S

S

S S

S

S

S

S

g

S√

2 (1+

e)

p 2 − g

p 2 + g

(a) (b)

Figure 6.14

2.In alternating current theory we often need to add sinusoidal waves of the
same frequency but different amplitude and phase, such as A 1 sin(ωt+α 1 ) and