Download Chapter 5 Check Point - Miami Beach Senior High School

P-BLTZMC05_585-642-hr
21-11-2008
12:53
Page 618
618 Chapter 5 Analytic Trigonometry
Preview Exercises
1
3
109. cos2 a sin-1 b
2
5
1
3
110. sin2 a cos-1 b
2
5
111. Use a right triangle to write sin 12 sin-1 x2 as an algebraic
expression. Assume that x is positive and in the domain of
the given inverse trigonometric function.
Exercises 113–115 will help you prepare for the material covered in
the next section. In each exercise, use exact values of trigonometric
functions to show that the statement is true. Notice that each statement
expresses the product of sines and/or cosines as a sum or a difference.
113. sin 60° sin 30° = 12 3cos 160° - 30°2 - cos 160° + 30°24
114. cos
6
112. Use the power-reducing formulas to rewrite sin x as an
equivalent expression that does not contain powers of
trigonometric functions greater than 1.
Chapter
p
p
1
p
p
p
p
cos = B cos a - b + cos a + b R
2
3
2
2
3
2
3
115. sin p cos
1
p
p
p
= B sin a p + b + sin ap - b R
2
2
2
2
Mid-Chapter Check Point
5
What you Know: Verifying an identity means showing that the expressions on each side are identical. Like solving
puzzles, the process can be intriguing because there are sometimes several “best” ways to proceed. We presented some
guidelines to help you get started (see page 593). We used fundamental trigonometric identities (see page 586), as well as
sum and difference formulas, double-angle formulas, power-reducing formulas, and half-angle formulas (see page 614) to
verify identities. We also used these formulas to find exact values of trigonometric functions.
Study Tip
Make copies of the boxes on pages 586 and 614 that contain the essential trigonometric identities. Mount these boxes on cardstock and
add this reference sheet to the one you prepared for Chapter 4. (If you didn’t prepare a reference sheet for Chapter 4, it’s not too late:
See the study tip on page 580.)
In Exercises 1–18, verify each identity.
1. cos x1tan x + cot x2 = csc x
2.
sin1x + p2
= tan2 x - sec2 x
3p
cos ¢ x +
≤
2
cos t - cot t
sin t - 1
4.
=
cos t
cos t cot t
7.
18.
sec t - 1
1 - cos t
=
t sec t
t
1 + csc x
10.
- cot x = cos x
sec x
cot x - 1
1 - tan x
11.
=
cot x + 1
1 + tan x
12. 2 sin3 u cos u + 2 sin u cos3 u = sin 2u
3
,
5
p
6 a 6 p
2
12
3p
.
cos b = - , p 6 b 6
13
2
csc u
t
tan t - sin t
=
2
2 tan t
sin t + cos t
sin t
=
sec t + csc t
sec t
1 - tan2 a tan2 b
1
2 tan x
=
csc 2x
1 + tan2 x
cos u11 + cot u2
1
9. sin a cos b = 3sin1a + b2 + sin1a - b24
2
13.
17.
tan2 a - tan2 b
sin u
1 - cos u
sin a =
sin x
cos x
+
= sin x + cos x
tan x
cot x
8. sin2
15. tan1a + b2 tan1a - b2 =
Use the following conditions to solve Exercises 19–22:
1 - cos 2x
= tan x
sin 2x
6. sin u cos u + cos2 u =
sec2 x
2 - sec2 x
16. csc u + cot u =
3. 1sin u + cos u22 + 1sin u - cos u22 = 2
5.
14. sec 2x =
Find the exact value of each of the following.
19. cos1a - b2
20. tan1a + b2
21. sin 2a
22. cos
b
2
In Exercises 23–26, find the exact value of each expression. Do not
use a calculator.
23. sin a
25. cos
5p
3p
+
b
4
6
5p
p
5p
p
cos
+ sin
sin
12
12
12
12
26. tan 22.5°
24. cos2 15° - sin2 15°