Download Sum and Difference Identities

EXAMPLE 1
Evaluate a trigonometric expression
Find the exact value of (a) sin 15° and (b) tan 7π .
12
Substitute 60° – 45°
a. sin 15° = sin (60° – 45°)
for 15°.
= sin 60° cos 45° – cos 60° sin 45° Difference
formula for sine
= 3 ( 2 )– 1 ( 2 )
2
2
2 2
=
6– 2
4
Evaluate.
Simplify.
EXAMPLE 1
Evaluate a trigonometric expression
b. tan 7π = tan ( π + π )
12
3
4
π
π
tan 3 + tan 4
=
π
π
1 – tan 3 tan 4
Substitute π + π For 7π .
3
12
4
Sum formula for tangent.
3 +1
=1– 3 1
Evaluate.
= –2 – 3
Simplify.
EXAMPLE 2
Use a difference formula
4
3π
Find cos (a – b) given that cos a = – 5 with π < a < 2 and
sin b = 5 with 0 < b < π .
13
2
SOLUTION
Using a Pythagorean identity and quadrant signs
gives sin a = – 3 and cos b = 12 .
13
5
cos (a – b) = cos a cos b + sin a sin b Difference formula for cosine
3 5
4 12
Substitute.
= – ( ) + (– )( )
5 13
5 13
63
Simplify.
= – 65
GUIDED PRACTICE
for Examples 1 and 2
Find the exact value of the expression.
1.
sin 105°
3.
ANSWER
6 +
4
2.
2+ 3
4.
ANSWER
6 –
4
ANSWER
2
cos 75°
2
tan 5π
12
cos π
12
ANSWER
2 +
4
6
GUIDED PRACTICE
5.
for Examples 1 and 2
8
π
Find sin (a – b) given that sin a = 17 with 0 < a < 2
3π
and cos b = – 24 with π < b < .
2
25
ANSWER
87
– 425
EXAMPLE 3
Simplify an expression
Simplify the expression cos (x + π).
cos (x + π) = cos x cos π – sin x sin π
Sum formula for cosine
= (cos x)(–1) – (sin x)(0)
Evaluate.
= – cos x
Simplify.
EXAMPLE 4
Solve a trigonometric equation
Solve sin ( x + π ) + sin ( x – π ) = 1 for 0 ≤ x < 2π.
3
3
Write equation.
sin ( x + π ) + sin ( x – π ) = 1
3
3
Use formulas.
sin x cos π + cos x sin π + sin x cos π – cos x sin π = 1
3
3
3
3
1
3
1
3
sin
x
+
cos
x
+
sin
x
–
2
2
2
2 cos x = 1 Evaluate.
sin x = 1 Simplify.
ANSWER
In the interval 0 ≤ x <2π, the only solution is x = π .
2
EXAMPLE 5
Solve a multi-step problem
Daylight Hours
The number h of hours of daylight for Dallas, Texas,
and Anchorage, Alaska, can be approximated by the
equations below, where t is the time in days and t = 0
represents January 1. On which days of the year will
the two cities have the same amount of daylight?
Dallas:
h1 = 2 sin ( π t – 1.35) + 12.1
182
Anchorage:
h2 = –6cos ( π t ) + 12.1
182
EXAMPLE 5
Solve a multi-step problem
SOLUTION
STEP 1
Solve the equation h1 = h2 for t.
2 sin ( π t – 1.35) + 12.1 = – 6 cos ( π t ) + 12.1
182
182
sin ( π t – 1.35) = – 3 cos ( π t )
182
182
sin ( π t ) cos 1.35 – cos ( π t ) sin 1.35 = – 3 cos ( π t )
182
182
182
sin ( π t ) (0.219) – cos ( π t ) (0.976) = – 3 cos ( π t )
182
182
182
0.219 sin ( π t ) = – 2.024 cos ( π t )
182
182
EXAMPLE 5
Solve a multi-step problem
tan ( π t ) = – 9.242
182
πt
–1 (– 9.242) + nπ
=
tan
182
πt
– 1.463 + nπ
182
t
STEP 2
– 84.76 + 182n
Find the days within one year (365 days)
for which Dallas and Anchorage will have
the same amount of daylight.
t
– 84.76 + 182(1)
97, or on April 8
t
– 84.76 + 182(2)
279, or on October 7
GUIDED PRACTICE
for Examples 3, 4, and 5
Simplify the expression.
6.
sin (x + 2π)
ANSWER
sin x
7. cos (x – 2π)
ANSWER
cos x
8.
tan (x – π)
ANSWER
tan x
GUIDED PRACTICE
for Examples 3, 4, and 5
9. Solve 6 cos ( π t ) + 5 = – 24 sin ( π t + 22) + 5 for 0 ≤ t < 2π.
75
75
ANSWER
about 5.65