Higher Engineering Mathematics, Sixth Edition

170 Higher Engineering Mathematics

LHS=

1 −cos2θ

sin2θ

=

1 −( 1 −2sin^2 θ)

2sinθcosθ

=

2sin^2 θ

2sinθcosθ

=

sinθ

cosθ

=tanθ=RHS

Problem 13. Prove that

cot 2x+cosec 2x=cotx.

LHS=cot2x+cosec 2x=

cos2x

sin2x

+

1

sin2x

=

cos2x+ 1

sin2x

=

(2cos^2 x− 1 )+ 1

sin2x

=

2cos^2 x

sin2x

=

2cos^2 x

2sinxcosx

=

cosx

sinx

=cotx=RHS

Problem 14. Solve the equation

cos2θ+3sinθ=2forθin the range 0◦≤θ≤ 360 ◦.

Replacing the double angle term with the relationship

cos2θ= 1 −2sin^2 θgives:

1 −2sin^2 θ+3sinθ= 2

Rearranging gives: −2sin^2 θ+3sinθ− 1 = 0

or 2sin^2 θ−3sinθ+ 1 = 0

which is a quadratic in sinθ

Using the quadratic formula or by factorising gives:

(2sinθ− 1 )(sinθ− 1 )= 0

from which, 2sinθ− 1 =0orsinθ− 1 = 0

and sinθ=^12 or sinθ= 1

from which, θ= 30 ◦or 150◦or 90◦

Now try the following exercise

Exercise 74 Further problems on double

angles

1. The powerpin an electrical circuit is given by

p=

v^2

R

. Determine the power in terms ofV,
Rand cos2twhenv=Vcos[t.
V^2
2 R

( 1 +cos2t)

]

2. Prove the following identities:
   (a) 1−

cos2φ

cos^2 φ

=tan^2 φ

(b)

1 +cos2t

sin^2 t

=2cot^2 t

(c)

(tan2x)( 1 +tanx)

tanx

=

2

1 −tanx

(d) 2cosec2θcos2θ=cotθ−tanθ

3. If the third harmonic of a waveform is given by
   V 3 cos3θ, express the third harmonic in terms
   of the first harmonic cosθ,whenV 3 =1.
   [cos3θ=4cos^3 θ−3cosθ]
   In Problems 4 to 8, solve for θ in the range
   − 180 ◦≤θ≤ 180 ◦


4. cos2θ=sinθ [− 90 ◦, 30 ◦, 150 ◦]


5. 3sin2θ+2cosθ= 0
   [− 160. 47 ◦,− 90 ◦,− 19. 47 ◦, 90 ◦]


6. sin2θ+cosθ= 0
   [− 150 ◦,− 90 ◦,− 30 ◦, 90 ◦]


7. cos2θ+2sinθ=−3[− 90 ◦]


8. tanθ+cotθ=2[ 45 ◦,− 135 ◦]

17.4 Changing products of sines and

cosines into sums or differences

(i) sin(A+B)+sin(A−B)=2sinAcosB(fromthe

formulae in Section 17.1)

i.e. sinAcosB

=^12 [sin(A+B)+sin(A−B)] (1)

(ii) sin(A+B)−sin(A−B)=2cosAsinB

i.e. cosAsinB

=^12 [sin(A+B)−sin(A−B)] (2)

(iii) cos(A+B)+cos(A−B)=2cosAcosB

i.e. cosAcosB

=^12 [cos(A+B)+cos(A−B)] (3)