Section 5.4, Sum and Difference Formulas
Homework: 5.4 #3, 5, 11, 13, 19, 21, 37, 45, 51, 55, 61
In Chapter 4, we looked at finding the trigonometric functions of many angles. In this section, we
will be able to find trigonometric functions for sums and differences of angles. This will allows for
angles not listed on our unit circles such as π/12 = π/4 − π/6.
The formulas that we will use are:
sin(u + v) = sin u cos v + cos u sin v
sin(u − v) = sin u cos v − cos u sin v
cos(u + v) = cos u cos v − sin u sin v
cos(u − v) = cos u cos v + sin u sin v
tan u + tan v
tan(u + v) =
1 − tan u tan v
tan u − tan v
tan(u − v) =
1 + tan u tan v
There are proofs of these on page 424 of your book. These were derived by a Greek astronomer
named Hipparchus, who was born in 160 B.C. He is considered the inventor of trigonometry.
Examples
1. Calculate each of the following:
π
(a) sin
12
We can use that π/12 = π/4 − π/6 to get that
π
π π
sin
= sin
−
12
4
6
π
π
π
π
= sin cos − cos sin
4
6
4
6
√ √
√
2
3
2 1
=
·
−
·
2
2
2
2
√
√
6− 2
=
4
7π
12
Note that 7π/12 = π/3 + π/4.
7π
π π
cos
= cos
+
12
3
4
π
π
π
π
= cos cos − sin sin
3
4
3
4
√
√ √
1
2
3 2
= ·
−
2
√ 2√ 2 2
2− 6
=
4
(b) cos
5π
12
We can use that 5π/12 = 3π/4 − π/3 (There are many other sums and differences of
(c) tan
angles that will work.)
5π
3π π
tan
= tan
−
12
4
3
π
tan 3π
4 − tan 3
3π
1 + tan 4 tan π3
√
−1 − 3
√
=
1 + (−1) 3
√
√
−1 − 3 1 + 3
√ ·
√
=
1− 3 1+ 3
√
√
−4 − 2 3
=
=2+ 3
−2
=
cos v = − 45 , and both u and v are in Quadrant II.
12
4
5
3
· −
− −
·
sin(u − v) = sin u cos v − cos u sin v =
13
5
13
5
−48 + 15
33
=
=− ,
65
65
2. Find sin(u − v) if sin u =
5
where − 13
and
3
5
12
13 ,
are from the Pythagorean identity, sin2 θ + cos2 θ = 1.
3. Confirm the identity sin(x + y) + sin(x − y) = 2 sin x cos y
sin(x + y) + sin(x − y) = sin x cos y + cos x sin y + sin x cos y − cos x sin y
= 2 sin x cos y
4. Verify that cos( π2 + x) = − sin x
π
π
π
+ x = cos cos x − sin sin x
cos
2
2
2
= − sin x