Download 1316-t4-review - Lone Star College

TEST 4 - REVIEW
Verify the identiry.
1) tan2 x = sec2 x - sin2 x - cos2 x
Answer: Identity
2)
sin x
sin x
+
= 2 csc x
1 - cos x 1 + cos x
Answer: Identity
3) cos2 x - sin2 x = 1 - 2 sin2 x
Answer: Identity
4)
cot2 x
1 + sin x
=
csc x - 1
sin x
Answer: Identity
5)
1 + csc x
= cos x + cot x
sec x
Answer: Identity
6)
sin3 x + sin x cos2 x
= cos x
tan x
Answer:
sin3 x + sin x cos2 x
sin x(sin2 x + cos2 x)
=
tan x
tan x
=
sin x · 1
tan x
Pythagorean identity
= sin x ÷
sin x
cos x
= sin x ·
cos x
sin x
= cos x
Prove the identity.
7) sec3 x cos x - sec2 x sin2 x = 1
Answer: sec3 x cos x - sec2 x sin2 x =
=
1
cos3 x
1
cos2 x
-
· cos x -
1
· sin2 x
2
cos x
sin2 x
cos2 x
= sec2 x - tan2 x
=1
Pythagorean identity
1
Find the exact value using a double-angle identity.
10
8) sin
3
3
2
Answer: -
Use an identity to write the expression as a single trigonometric function or as a single number.
2 tan 15°
9)
1 - tan2 15°
Answer:
3
3
10) 2 cos2 22.5° - 1
Answer:
2
2
Find the exact value by using a half-angle identity.
11) sin 22.5°
Answer:
1
2
2- 2
Simplify the expression by using a sum or difference identity.
12) cos(14k) cos (4k) - sin(14k) sin (4k)
Answer: cos(18k)
Use trigonometric identities to find the exact value.
tan 43° + tan 17°
13)
1 - tan 43° tan 17°
Answer:
3
14) sin 10° cos 50° + cos 10° sin 50°
Answer:
3
2
Use an appropriate identity to find the exact value of the expression.
15) cos (75°)
Answer:
6- 2
4
Find the exact value by using a sum or difference identity.
16) tan 105°
Answer: -2 - 3
2
Find the exact value of the expression using the provided information.
12
15
, with A in quadrant II, and sin B =
, with B in quadrant II.
17) Find cos(A - B) given that cos A = 13
17
Answer:
171
221
Find the exact value of the composition.
1
18) cos arcsin
4
Answer:
15
4
Evaluate the expression exactly.
19) csc(csc-1 3)
Answer:
3
Solve the equation exactly over the interval [0, 2 ).
1
20) sin x cos x =
2
Answer:
5
4 4
,
Find all real numbers in the interval [0, 2 ) that satisfy the equation.
21) sin2 x - cos2 x = 0
Answer:
3
5
7
,
,
4 4
4
4
,
Find all real numbers in [0, 2 ] that satisfy the equation.
3
22) sin 4x =
2
Answer:
2
7
7
13 5
19
,
,
,
,
,
12 6 3 12 6
12 3
12
,
,
Find all values of x in the interval [0°, 360°) that satisfy the equation. Round approximate answers to the nearest
tenth of a degree.
tan 4x - tan x
3
=
23)
1 + tan 4x tan x
3
Answer: {10°, 70°, 130°, 190°, 250°, 310°}
24) 7 cot2 x - 5 = 0
Answer: {49.8°, 130.2°, 229.8°, 310.2°}
Find all values of
in [0°, 360°) that satisfy the equation.
25) 2 sin 2 - 3 = 0
Answer: {30°, 60°, 210°, 240°}
3