164 Higher Engineering Mathematics
tan
x +
π
4
=
tan x + tan
π
4
1 − tan x tan
π
4
from the formula fortan(A + B)
=
tan x + 1
1 − (tan x)(1)
=
1 + tan x
1 − tan x
since tan
π
4
= 1
tan
x −
π
4
=
tan x − tan
π
4
1 + tan x tan
π
4
=
tan x − 1
1 + tan x
Hence tan
x +
π
4
tan
x −
π
4
=
1 + tan x
1 − tan x
tan x − 1
1 + tan x
=
tan x − 1
1 − tan x
=
−(1 − tan x)
1 − tan x
= −1
Problem 4. If sin P = 0.8142 and cos Q = 0.4432
evaluate, correct to 3 decimal places:
(a) sin(P − Q), (b) cos(P + Q) and
(c) tan(P + Q), using the compound-angle
formulae.
Since sin P = 0.8142 then
P = sin
−1 0.8142 =54.51 ◦ .
Thus cos P = cos 54.51 ◦ = 0.5806 and
tan P = tan 54.51 ◦ = 1.4025
Since cos Q = 0.4432, Q = cos −1 0.4432 =63.69 ◦ .
Thus sin Q = sin 63.69 ◦ = 0.8964 and
tan Q = tan 63.69 ◦ = 2.0225
(a) sin(P − Q)
= sin P cos Q − cos P sin Q
= (0.8142)(0.4432) − (0.5806)(0.8964)
= 0.3609 − 0.5204 = −0.160
(b) cos(P + Q)
= cos P cos Q − sin P sin Q
= (0.5806)(0.4432) − (0.8142)(0.8964)
= 0.2573 − 0.7298 = −0.473
(c) tan(P + Q)
=
tan P + tan Q
1 − tan P tan Q
=
(1.4025) + (2.0225)
1 − (1.4025)(2.0225)
=
3.4250
−1.8366
= −1.865
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 forsin(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 sin39.449
◦
= 2.542
RHS = 5 cos x = 5 cos59.449
◦
= 2.542]
Now try the following exercise
Exercise 72 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 ◦ ≡ −cos 76 ◦
(b)cos
π
12
⎤
⎦