# Double-Angle And Half-Angle Identities Worksheets With Answer Page 8

478
6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS
In Problems 11–14, graph y
and y
in the same viewing window
Suppose you are tutoring a student who is having difﬁculties in
1
2
for
2
x
2 . Use TRACE to compare the two graphs.
ﬁnding the exact values of sin and cos from the information
given in Problems 41 and 42. Assuming you have worked
2
2
11. y
cos 2x, y
cos
x
sin
x
through each problem and have identiﬁed the key steps in the
1
2
solution process, proceed with your tutoring by guiding the
12. y
sin 2x, y
2 sin x cos x
1
2
student through the solution process using the following
x
sin x
questions. Record the expected correct responses from the
13.
y
tan
, y
1
2
2
1
cos x
student.
(A) The angle 2 is in what quadrant and how do you know?
2 tan x
14.
y
tan 2x, y
(B) How can you ﬁnd sin 2 and cos 2 ? Find each.
1
2
2
1
tan
x
(C) What identities relate sin and cos with either sin 2 or
cos 2 ?
B
(D) How would you use the identities in part C to ﬁnd sin
and cos exactly, including the correct sign?
Verify the identities in Problems 15–32.
(E) What are the exact values for sin and cos ?
2
15. (sin x
cos x)
1
sin 2x
41.
Find the exact values of sin and cos , given tan 2
4
, 0°
90°.
16. sin 2x
(tan x)(1
cos 2x)
3
1
1
2
2
17. sin
x
(1
cos 2x)
18. cos
x
(cos 2x
1)
42.
Find the exact values of sin and cos , given sec 2
2
2
5
, 0°
90°.
4
19. 1
cos 2x
tan x sin 2x
2
20. 1
sin 2t
(sin t
cos t)
Verify each of the following identities for the value of x
indicated in Problems 43–46. Compute values to ﬁve
x
1
cos x
x
1
cos x
2
2
21.
sin
22.
cos
signiﬁcant digits using a calculator.
2
2
2
2
2 tan x
x
1
cos x
(A)
tan 2x
(B)
cos
2
1
tan
x
cot x
tan x
2
1
tan
x
2
2
23.
cot 2x
24.
cot 2x
2 tan x
2
(Choose the correct sign.)
sin
1
cos
43. x
252.06°
44. x
72.358°
25.
cot
26.
cot
2
1
cos
2
sin
45. x
0.934 57
46. x
4
2
1
tan
u
cos 2u
1
tan u
27.
cos 2u
28.
2
1
tan
u
1
sin 2u
1
tan u
In Problems 47–50, graph y
and y
in the same viewing
1
2
window for
2
x
2 , and state the intervals for which
2
2
1
tan
x
sec
x
29.
2 csc 2x
30.
sec 2x
the equation y
y
is an identity.
2
tan x
2
sec
x
1
2
1
cos x
2
1
tan
( /2)
cot
tan
47.
y
cos (x/2), y
31.
cos
32.
cos 2
1
2
2
2
1
tan
( /2)
cot
tan
1
cos x
Compute the exact values of sin 2x, cos 2x, and tan 2x using
48.
y
cos (x/2), y
1
2
2
the information given in Problems 33–36 and appropriate
identities. Do not use a calculator.
1
cos x
49.
y
sin (x/2), y
1
2
2
3
4
33.
sin x
, /2
x
34.
cos x
, /2
x
5
5
1
cos x
5
35.
tan x
,
/2
x
0
50.
y
sin (x/2), y
12
1
2
2
5
36.
cot x
,
/2
x
0
12
C
In Problems 37–40, compute the exact values of sin (x/2),
cos (x/2), and tan (x/2) using the information given and
appropriate identities. Do not use a calculator.
Verify the identities in Problems 51–54.
1
3
37.
sin x
,
x
3 /2
51. cos 3x
4 cos
x
3 cos x
3
1
3
38.
cos x
,
x
3 /2
52. sin 3x
3 sin x
4 sin
x
4
3
4
2
39.
cot x
,
x
/2
53. cos 4x
8 cos
x
8 cos
x
1
4
3
3
40. tan x
,
x
/2
54. sin 4x
(cos x)(4 sin x
8 sin
x)
4