ABSTRACT
Problem 5. Solve the equation: 4 sin(x − 20◦) = 5 cos x for values of x between 0◦ and 90◦
4 sin (x − 20◦) = 4[sin x cos 20◦ − cos x sin 20◦] from the formula for sin (A − B)
= 4[sin x(0.9397) − cos x(0.3420)] = 3.7588 sin x − 1.3680 cos x
Since 4 sin(x − 20◦) = 5 cos x then 3.7588 sin x − 1.3680 cos x = 5 cos x Rearranging gives:
3.7588 sin x = 5 cos x + 1.3680 cos x = 6.3680 cos x
and sin x cos x
= 6.3680 3.7588
= 1.6942
i.e. tan x = 1.6942, and x = tan−1 1.6942 = 59.449◦ or 59◦27′
[Check: LHS = 4 sin(59.449◦ − 20◦) = 4 sin 39.449◦ = 2.542
RHS = 5 cos x = 5 cos 59.449◦ = 2.542]
Now try the following exercise
Exercise 104 Further problems on compound angle formulae
1. Reduce the following to the sine of one angle: (a) sin 37◦ cos 21◦ + cos 37◦ sin 21◦ (b) sin 7t cos 3t − cos 7t sin 3t
[(a) sin 58◦ (b) sin 4t] 2. Reduce the following to the cosine of one
angle: (a) cos 71◦ cos 33◦ − sin 71◦ sin 33◦
(b) cos π 3
cos π
4 + sin π
3 sin
π
4⎡ ⎣ (a) cos 104
◦ ≡ −cos76◦ (b) cos π
⎤ ⎦
3. Show that: (a) sin
( x + π
) + sin
( x + 2π
) = √
3 cos x
(b) −sin (
3π 2
−φ )
= cos φ 4. Prove that:
(a) sin ( θ + π
) − sin
( θ − 3π
) =
√ 2(sin θ + cos θ)
(b) cos (270 ◦ + θ)
cos (360◦ − θ) = tan θ 5. Given cos A = 0.42 and sin B = 0.73 evaluate
(a) sin(A − B), (b) cos(A − B), (c) tan(A + B), correct to 4 decimal places.
