Higher Engineering Mathematics

184 GEOMETRY AND TRIGONOMETRY

i.e. instantaneous power,

p=25[cosπ/ 6 −cos (2ωt−π/6)]

Now try the following exercise.

Exercise 83 Further problems on changing

products of sines and cosines into sums or

differences

In Problems 1 to 5, express as sums or differ-

ences:

1. sin 7tcos 2t

[ 1

2 (sin 9t+sin 5t)

]

2. cos 8xsin 2x

[ 1

2 (sin 10x−sin 6x)

]

3. 2 sin 7tsin 3t [cos 4t−cos 10t]


4. 4 cos 3θcosθ [2(cos 4θ+cos 2θ)]


5. 3 sin

π

3

cos

π

6

[

3

2

(

sin

π

2

+sin

π

6

)]

6. Determine

∫

2 sin 3t[costdt

−

cos 4t

4

−

cos 2t

2

+c

]

7. Evaluate

∫ π

2

0

4 cos 5xcos 2xdx

[

−

20

21

]

8. Solve the equation: 2 sin 2φsinφ=cosφin
   the rangeφ=0toφ= 180 ◦.
   [30◦,90◦or 150◦]

18.5 Changing sums or differences of

sines and cosines into products

In the compound-angle formula let,

(A+B)=X

and

(A−B)=Y

Solving the simultaneous equations gives:

A=

X+Y

2

andB=

X−Y

2

Thus sin(A+B)+sin(A−B)=2 sinAcosB
becomes,

sinX+sinY

=2 sin

(

X+Y

2

)

cos

(

X−Y

2

)

(5)

Similarly,

sinX−sinY

=2 cos

(

X+Y

2

)

sin

(

X−Y

2

)

(6)

cosX+cosY

=2 cos

(

X+Y

2

)

cos

(

X−Y

2

)

(7)

cosX−cosY

=−2 sin

(

X+Y

2

)

sin

(

X−Y

2

)

(8)

Problem 18. Express sin 5θ+sin 3θ as a

product.

From equation (5),

sin 5θ+sin 3θ=2 sin

(

5 θ+ 3 θ

2

)

cos

(

5 θ− 3 θ

2

)

=2 sin 4θcosθ

Problem 19. Express sin 7x−sinx as a

product.

From equation (6),

sin 7x−sinx=2 cos

(

7 x+x

2

)

sin

(

7 x−x

2

)

=2 cos 4xsin 3x

Problem 20. Express cos 2t−cos 5t as a

product.

From equation (8),

cos 2t−cos 5t=−2 sin

(

2 t+ 5 t

2

)

sin

(

2 t− 5 t

2

)

=−2 sin

7

2

tsin

(

−

3

2

t

)

=2 sin

7

2

tsin

3

2

t

(

since sin

(

−

3

2

t

)

=−sin

3

2

t

)

Problem 21. Show that

cos 6x+cos 2x

sin 6x+sin 2x

=cot 4x.