Compound angles 171
(iv) cos(A + B) − cos(A − B) = −2 sin A sin B
i.e. sin A sin B
=−
1
2 [cos(A + B) − cos(A − B)]
(4)
Problem 15. Express sin 4x cos3x as a sum or
difference of sines and cosines.
From equation (1),
sin 4x cos 3x =
1
2 [sin(4x + 3x) + sin(4x − 3x)]
=
1
2 (sin 7x + sin x)
Problem 16. Express 2 cos 5θ sin 2θ as a sum or
difference of sines or cosines.
From equation (2),
2 cos5θ sin 2θ = 2
1
2
[sin(5θ + 2θ) − sin(5θ −2θ)]
= sin 7θ − sin 3θ
Problem 17. Express 3 cos4t cos t as a sum or
difference of sines or cosines.
From equation (3),
3 cos4t cos t = 3
1
2
[cos(4t + t ) + cos(4t − t )]
=
3
2
(cos 5t + cos 3t)
Thus, if the integral
3 cos4t cos t dt was required (for
integration see Chapter 37), then
3 cos4t cos t dt =
3
2
(cos 5t + cos 3t ) dt
=
3
2
sin 5t
5
+
sin 3t
3
+ c
Problem 18. In an alternating current circuit,
voltage v = 5 sinωt and current i = 10 sin(ωt −
π/6). Find an expression for the instantaneous
power p at time t given that p = vi, expressing the
answer as a sum or difference of sines and cosines.
p = vi = (5 sin ωt ) [10 sin (ωt − π/6)]
= 50 sin ωt sin(ωt − π/6)
From equation (4),
50 sin ωt sin(ωt − π/6)
= (50)
−
1
2
cos(ωt + ωt − π/6)
− cos
ωt − (ωt − π/6)
= −25{cos(2ωt − π/6) − cos π/6}
i.e. instantaneous power,
p = 25[cos π /6 − cos (2ω t − π/6)]
Now try the following exercise
Exercise 75 Further problems on changing
products of sines and cosines into sums or
differences
In Problems 1 to 5, express as sums or differences:
1. sin 7t cos 2t
1
2 (sin 9t + sin 5t )
2. cos 8x sin 2x
1
2 (sin 10x − sin 6x)
3. 2 sin 7t sin 3t
[cos4t − cos 10t ]
4. 4 cos3θ cos θ
[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 cos t dt .
−
cos 4t
4
−
cos 2t
2
+ c
7. Evaluate
π
2
0
4 cos 5x cos 2x dx.
−
20
21
8. Solve the equation: 2 sin 2φ sin φ = cos φ in
the range φ = 0 to φ = 180 ◦ .
[30 ◦ , 90 ◦ or 150 ◦ ]
17.5 Changing sums or differences of
sines and cosines into products
In the compound-angle formula let,
(A + B) = X
and
(A − B) = Y