Download Study Guide Pre Calc 5.4

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LESSON 5.4 STUDY GUIDE
GOAL:
Use trigonometric sum and difference formulas.
EXAMPLE 1: Evaluate a trigonometric expression
Find the exact value of cos 75
Solution
cos 75  (45  30)
 cos 45 cos 30  sin 45 sin 30
2  3 
2 1


 


2  2  2  2

6 2
4
Substitute 45° + 30° for 75°.
Sum formula for cosine
Evaluate.
Simplify.
Exercises for Example 1: Using the Sum and Difference Formulas
Find the exact value of the expression. (answer in radical form not decimal form)
1. cos 15
2. tan 15
3. sin 75
4. cos 105
5. sin

12
6. cos
5
12
7. tan
13
12
8.
7
12
sin
EXAMPLE 2:
Find sin(a  b) given that sin a 
4
π
12
3π
with 0 < a < and cos b   With  < b <
5
2
2
 13
Solution
Using a Pythagorean identity and quadrant signs gives cos a  3 and sin b  5 .
5
13
sin (a  b)  sin a cos b  cos a sin b

4  12  3  5 
    
5  14  5  13 

33
65
Difference formula for sine
Substitute.
Simplify.
Exercises for Example 2

1
Evaluate the expression given that cos a  with < a <  and sin b  2
3
2
3
π
with 0 < b < .
2
9. sin(a  b)
10. cos(a  b)
11. tan(a  b)
12. sin(a  b)
EXAMPLE 3: Simplify an expression
Simplify the expression sin  x  π.
 2
Solution
sin  x  π   sin x cos   cos x sin 
2
2
 2
 (sin x)(0)  (cos x)(l)
 cos x
Sum formula for sine
Evaluate.
Simplify.
Exercises for Example 3
Simplify the expression.
13. cos x  π 

2

14. tan(x  2)
EXAMPLE 4: Solve a trigonometric equation
Solve: sin (x  )  sin (x  )  1 for 0  x < 2.
Solution
sin(x  )  sin(x  )  1
Write equation.
sin x cos   cos x sin   sin x cos   cos x sin   1
Use formulas.
(sin x) (1)  (cos x) (0)  (sin x) (1)  (cos x) (0)  1
Evaluate.
2 sin x  1
Simplify.
5π
 1  π
In the interval 0  x < 2, the solutions of x  sin1  are and .
6
6
 2
Exercise for Example 4
16. Solve sin x  π   sin  x  π  = 1 for 0 x <2.
4
4

