Download Review TEST 2 MAC 1114 Sum B

Review TEST 2 MAC 1114 Sum B
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Solve the problem.
1) A wheel with a 23-inch diameter is turning at the rate of 47 revolutions per minute. To the nearest inch per
minute, what is the velocity of a point on the rim?
Answer: 3396 in./min
Graph the function over a one-period interval.
1
2) y = 2 + sin (2x - )
3
Answer:
Find the exact value of the expression in degrees without using a calculator or table.
2
3) = cos-1
2
Answer: 45°
4)
= csc-1 (-2)
Answer: -30°
5)
= sec-1 -
2
Answer: 135°
1
Find the exact value of the composition.
6) sin (arctan (2))
Answer:
2 5
5
7) cos arcsin
1
4
Answer:
15
4
3
8) tan-1 tan
4
Answer: -
4
Use a calculator to find the approximate value of the composition. Round answers to four decimal places. The expression
may be undefined.
9) sin (cos-1 (0.8324))
Answer: 0.5542
Find the exact value of the indicated trigonometric function for the given right triangle.
10)
20
12
Find sin A and cos A.
Answer: sin A =
4
3
; cos A =
5
5
Solve the right triangle with the given sides and angles.
11) a = 3.8 cm, b = 1.4 cm
Answer:
= 69.8°,
= 20.2°, c = 4.0 cm
Perform the computation and give answer with the appropriate number of significant digits.
62
12)
sin (21.597°)
Answer: 170
Solve the problem.
13) From a boat on the lake, the angle of elevation to the top of a cliff is 33°7'. If the base of the cliff is 986 feet from
the boat, how high is the cliff (to the nearest foot)?
Answer: 643 ft
2
14) From a distance of 48 feet from the base of a building, the angle of elevation to the top of the building is 69°.
Estimate the height of the building to the nearest foot.
Answer: 125 ft
15) A blimp is 1700 meters high in the air and measures the angles of depression to two stadiums to the west of the
blimp. If those measurements are 72.4° and 27.8°, how far apart are the two stadiums to the nearest meter?
Answer: 2685 m
Write the expression in terms of sines and/or cosines, and then simplify.
16) tan x csc x
Answer: sec x
17)
tan
sec
Answer: sin
18)
1
cos2
-
1
cot2
Answer: 1
19)
tan x
+ 1 (sin x - 1)
sec x
Answer: -cos2 x
20)
sec x csc x
tan x cot x
Answer:
1
sin x cos x
Express the given trigonometric function in terms of the indicated function.
21) sin in terms of csc
Answer:
22) csc
1
csc
in terms of tan
Answer:
± 1 + tan2
tan
Use identities to find the exact value of the trigonometric function.
2
< <2 .
23) Find sin , given that cos = and
5
2
Answer: -
21
5
3
24) Find sin , given that cos
Answer:
=
1
and 0 <
5
<
2
.
2 6
5
25) Find tan
Answer: -
26) Find sec
if sec
26
and
<
5
2
=
<2 .
1
5
if cos
=
1
and 0 <
6
<
2
.
Answer: 6
27) Find sin , given that sec
Answer: -
=
9 77
and - <
77
2
< 0.
2
9
Use odd and even identities to simplify the expression.
28) csc x - csc (-x)
Answer: 2 csc x
29) cos (-x) tan (-x)
Answer: - sin x
Determine whether the function is odd, even, or neither.
30) f(x) = sec(x2 )
Answer: Even
31) f(x) = x + cos x
Answer: Neither
32) f(x) = x csc x
Answer: Even
Use identities to simplify the expression.
cos2
+ csc sin
33)
sin2
Answer: csc2
34)
1
+ sec w cos w
cot2 w
Answer: sec2 w
4
35)
-1
sin x - csc x
Answer: tan x sec x
36) cos4 x + sin2 x cos2 x
Answer: cos2 x
Prove that the equation is an identity.
sin3 x + sin x cos2 x
= cos x
37)
tan x
Answer:
sin3 x + sin x cos2 x
sin x(sin2 x + cos2 x)
=
tan x
tan x
=
sin x · 1
tan x
= sin x ÷
sin x
cos x
= sin x ·
cos x
sin x
Pythagorean identity
= cos x
38)
1
1 - cos x
=
csc x + cot x
sin x
Answer:
1
1
csc x - cot x
=
·
csc x + cot x
csc x + cot x csc x - cot x
=
csc x - cot x
csc2 x - cot2 x
=
csc x - cot x
1
Pythagorean identity
= csc x - cot x
1
cos x
=
sin x sin x
=
39)
1 - cos x
sin x
sec2 x - tan2 (-x)
= sec x + tan x
sec x + tan(-x)
Answer:
sec2 x - tan2 (-x)
sec2 x - tan(-x) · tan(-x)
=
sec x + tan(-x)
sec x + tan(-x)
=
sec2 x - (- tan x) · (-tan x)
sec x + [- tan x]
=
sec2 x - tan2 x
sec x - tan x
=
(sec x - tan x)(sec x + tan x)
sec x - tan x
= sec x + tan x
5
(tan(-x) = - tan x)
40)
9 csc2x - 19 csc x - 24
9
=
+8
csc x - 3
sin x
Answer:
9 csc2x - 19 csc x - 24 (9 csc x + 8)(csc x - 3)
=
csc x - 3
csc x - 3
= 9 csc x + 8
9
=
+8
sin x
41)
csc x cot x
sin x
=
cot x csc x
cot x
Answer:
csc x cot x csc x · csc x - cot x · cot x
=
cot x csc x
cot x csc x
=
=
42) cos4 x =
csc2x - cot2 x
cot x csc x
1
cot x csc x
=
1
1
·
cot x csc x
=
1
· sin x
cot x
=
sin x
cot x
Pythagorean identity
1 - tan2 x + sin2 x tan2 x
sec2 x
Answer:
1 - tan2x + sin2x tan2 x
=
sec2 x
1-
= cos2 x · 1 -
sin2 x
sin2 x
+ sin2 x ·
2
cos x
cos2 x
1
cos2 x
sin2 x
sin2 x
+ sin2x ·
cos2 x
cos2x
= cos2 x - sin2 x + sin4x
= cos2 x - sin2 x(1 - sin2x)
= cos2 x - sin2 x · cos2x
= cos2 x(1 - sin2x)
Pythagorean identity
= cos2x · cos2 x
= cos4 x
Pythagorean identity
6
43)
sin
1 + cos
+
1 + cos
sin
Answer:
sin
1 + cos
= 2 csc
+
1 + cos
sin
=
=
=
44)
sin2
+ 1 + 2 cos + cos2
(1 + cos )(sin )
2(1 + cos )
= 2 csc .
(1 + cos )(sin )
cos x
sec x - 1
=
cos x + 1
tan2x
Answer:
sec x - 1
=
tan2x
1
-1
cos x
sin2 x
cos2 x
cos2 x
=
=
45)
sin2 + (1 + cos )2
(1 + cos )(sin )
1
-1
cos x
cos2x ·
sin2x
cos2x
cos x - cos2 x
sin2 x
=
cos x(1 - cos x)
1 - cos2 x
=
cos x(1 - cos x)
(1 - cos x)(1 + cos x)
=
cos x
(1 + cos x)
Pythagorean identity
sec - 1
tan
=
tan
sec + 1
Answer:
sec - 1 (sec - 1)(sec + 1)
=
tan
(tan )(sec + 1)
=
(sec2 - 1)
(tan )(sec + 1)
=
tan2
(tan )(sec
+ 1)
tan
=
.
sec + 1
7
46) 1 -
cos2
1 + sin
Answer: 1 -
= sin
cos2
1 + sin
=
1 + sin - cos2
1 + sin
=
1 + sin + sin2 - 1
1 + sin
=
(sin )(1 + sin )
1 + sin
= sin .
47) (cos
- sin )2 + (cos
Answer: (cos
+ sin )2 = 2
- sin )2 + (cos
+ sin )2 = cos2
- 2 cos
8
sin
+ sin2
+ cos2
+ 2 cos
sin
+ sin2
= 2.