Download Trigonometry Review for Test 4 Name______________________

Trigonometry Review for Test 4
Name______________________
Name the quadrant that contains the terminal point of the given arc with initial point (1, 0) on the unit
circle.
1)
7
Answer: III
Find the exact functional value for arc x.
2) sin
4
Answer:
2
2
Find the exact functional value.
3
3) sin
4
Answer:
2
2
Determine the quadrant that contains the terminal point of t.
4) cos t < 0 and sin t > 0
Answer: II
Find the indicated functional value.
2
5) If cos t = and sin t > 0, find sin t.
3
Answer:
5
3
Find the reference arc for the given arc.
11
6)
4
Answer:
4
1
Prove that the equation is an identity.
7) cos x csc x tan x = 1
Answer: Answers may vary. One possibility:
cos t csc t tan t = 1
1
sin t
cos t
=1
sint cos t
1=1
8) tan2x = sec2x - sin2x - cos2x
Answer: Answers may vary. One possibility:
tan2 = sec2 - sin2 - cos2
sec2 - 1 = sec2 - sin2 - cos2
sec2
sec2
9)
- (sin2 + cos2 ) == sec2 - sin2 - cos2
- sin2 - cos2 = sec2 - sin2 - cos2
cot2x
1 + sin x
=
csc x - 1
sin x
Answer: Answers may vary. One possibility:
cot2
1 + sin
=
csc - 1
sin
csc2 - 1 1 + sin
=
csc - 1
sin
(csc
- 1)(csc
csc -1
csc
+1=
1
sin
+
sin
sin
1 + sin
sin
=
+ 1)
=
1 + sin
sin
1 + sin
sin
=
1 + sin
sin
1 + sin
sin
Find the exact functional value using the cosine sum or difference identity.
10) cos
12
Answer:
6+ 2
4
2
11) cos 165°
6+ 2
4
Answer: -
Find the exact functional value.
4
12) If sin A = - , with A in QIV, then find sin 2A.
5
Answer: -
24
25
Prove the identity.
13) sin x +
2
= cos x
Answer: Answers may vary. One possibility:
sin x +
= cos x
2
sin x cos
2
+ cos x sin
2
= cos x
(sin x)(0) + (cos x)(1) = cos x
cos x = cos x
Solve the equation for exact values of x, 0
14) sin x = 1 - 2 sin2x
Answer: x =
x<2 .
5 3
,
6 6 2
,
Solve the equation for the exact values of , 0°
15) 4 sin2 = 3
< 360°.
Answer: 60°, 120°, 240°, 300°
Solve the equation for , 0°
< 360°. When necessary, approximate solution(s) to the nearest tenth of a
degree.
16) sin2 + 8 sin + 16 = 0
Answer: No solution
Solve the equation for the interval [0, 2 ).
17) 2 sin2x = sin x
Answer: 0, ,
5
6 6
,
3
18) cos2x + 2 cos x + 1 = 0
Answer:
Solve the equation for exact values of x, 0°
19) sin 2x = cos x
x < 360°.
Answer: 30°, 90°, 150°, 270°
Find all solutions to the given equation.
20) cos x + sin x cos x = 0
Answer:
2
+k
Find the exact functional value.
7
and sin B < 0, then find sin 2B.
21) If tan B =
24
Answer:
336
625
Solve the problem.
1
1
and sin B = - , with A in QI and B in QIV, then find sin(A - B).
3
2
22) If cos A =
Answer:
2 6+1
6
Find the exact functional value using either a sine sum or difference identity or a tangent sum or
difference identity.
23) sin 255°
Answer: -
6+ 2
4
24) tan 165°
Answer:
3-2
Find the exact functional value.
24
, and 90° < 2 < 180°, then find sin .
25) If cos 2 = 25
Answer:
7 2
10
4
26) If tan x =
Answer:
7
, and 180° < x < 270°, then find tan 2x.
24
336
527
Find cos(A + B).
27) cos A = -
Answer:
1
4
and tan B = , with A and B in QIII.
6
3
3 - 4 35
30
Use a half-angle identity to find the exact value of the expression.
5
28) sin
12
Answer:
2+ 3
2
Convert the angle to radians. Leave as a multiple of .
29) 210°
Answer:
7
6
Convert the radian measure to degree measure. Round the answer to two decimal places.
7
30)
6
Answer: 210°
Find the angle of smallest possible positive measure that is coterminal with the given angle.
31) -49°
Answer: 311°
32) 472°
Answer: 112°
On a circle with the given radius r, find the length s of the arc intercepted by the central angle .
33) r = 10.18 ft,
=
30
(Round to the nearest tenth if necessary.)
Answer: 1.1 ft.
5
34) r = 36.14 in.,
=170° (Round to the nearest tenth if necessary.)
Answer: 107.2 in.
Find the radius r of a circle with central angle
35) = 1.5 radians, s = 9 in.
and arc length s.
Answer: 6 in.
Find the central angle of a circle with radius r and arc length s.
36) r = 10 cm, s = 25 cm
Answer: 2.5
Find the area of a sector with the given central angle
37)
=
2
,
in a circle of radius r.
r = 8 cm
Answer: 16 sq cm
Find sin , cos , or tan , as specified, for an angle
terminal side.
38) (9, 12)
Answer:
in standard position if the given point is on its
4
5
Solve the problem.
39) From a balloon 1133 feet high, the angle of depression to the ranger headquarters is 46°27'.
How far is the headquarters from a point on the ground directly below the balloon (to the
nearest foot)?
Answer: 1077 ft
Find the exact function value, if defined.
40) tan 30°
Answer:
3
3
41) sec 45°
Answer:
2
42) cos 210°
Answer: -
3
2
6
43) sin
5
3
Answer: -
44) sec
3
2
3
4
Answer: - 2
Solve the right triangle with the given sides and angles.
45) a = 3.8, b = 1.3
Answer:
= 71.1°,
= 18.9°, c = 4.0
Find the missing parts of the triangle.
46) B = 63°30'
a = 12.20 ft
c = 7.80 ft
Answer: b = 11.17 ft, A = 77°49', C = 38°41'
Find the missing parts of the triangle. (Find angles to the nearest hundredth of a degree.)
47) a = 160 yd
b = 188 yd
c = 325 yd
Answer: A = 19.25°, B = 22.79°, C = 137.96°
Use Heron's Formula to find the area of a triangle with sides a, b, and c.
48) a = 240
b = 123
c = 305
Answer: 13,860.45 sq units
49) Find the area of the triangle. Round to the nearest tenth.
ang;e B = 62° c = 7 a = 10
Answer: 30.9 in2
Sketch the graph of the given function on -2
x
2 . State the range and the x-intercepts.
7
50) y = 3 sin x
Answer:
Range: y 3
x-intercepts: x = k
8
Sketch the graph of the function for at least one period. Indicate the amplitude, period, phase shift (if
any), vertical shift (if any) and range.
51) y = 3cos (2x) + 3
Answer:
Amplitude: 3
Period:
Vertical shift: 3 up
Range: 0 y 6
52) y = sin(x - )
Answer:
Amplitude: 1
Period: 2
Phase shift: to the right
Range: y 1
Find the domain, range, and period of the given function.
53) y = sin x
Answer: D: x
R: y 1
Period: 2
9
54) y = cos x
Answer: D: x
R: y 1
Period: 2
55) y = tan x
Answer: D: x
2
+k
R: y
Period:
Find the exact value of the real number y.
1
56) y = arcsin
2
Answer:
6
2
2
57) y = sin-1 Answer: -
58) y = cos-1
Answer:
4
3
2
6
10
Graph the function on the indicated interval, using a solid line. Then graph the inverse, using a dashed
line.
x
59) y = sin x,
2
2
Answer:
11
60) y = cos x, 0
x
2
Answer:
61) Write an equation fpr the cosine function using the given informaion.
Amplitude = 7 Period = 3 Phase shift = left
Answer: y = ±7cos
2
3
3
62) The angle of elevation from the end of the shadow to the top of the building is 63° and the
distance is 220 feet. Find the height of the building to the nearest foot.
Answer: 196 feet
12
Without using a table or graphing utility, sketch the graph of the given function on -2
the range and the x-intercepts.
63) y = - 3 cos x
Answer:
Range: y
3
x-intercepts: x =
2
+k
Indicate the period and the range of the given function.
64) y = -5sec x
Answer: P = 2 ; Range: y
5
13
x
2 . State
Consider the given point graphed on the unit circle. Match the point that corresponds to it on the given
graph.
65)
Graph of y = sinx
A) A
B) D
C) C
D) B
Answer: C
14
66)
Graph of y = sinx
A) D
B) A
C) B
D) C
Answer: D
Find the exact cosine and sine values for the indicated arc on the unit circle.
67)
7
6
Answer: -
3 1
,2
2
15
Name the arc that describes t, the direction and length of the arc on the unit circle.
68)
A) B)
8
C) D)
4
8
4
Answer: D
16
69)
A)
B)
4
8
C) D) -
8
4
Answer: D
17
Find polar coordinates of the given point.
70)
A) 5, B) 5,
2
3
C) 5,
1
3
D) 5,
1
2
2
3
Answer: B
18
71)
A) 1.5, B) 2, -
3
4
C) 2.5,
3
4
D) 2.5, -
3
4
1
4
Answer: D
Convert to rectangular coordinates.
72) 5,
2
Answer: (0, 5)
Convert to polar coordinates. Express the answer in radians.
73) (12, -12)
Answer: 12 2,
7
4
74) (-3, -3 3)
Answer: 6,
4
3
Convert to a polar equation.
75) x2 - y2 = 4
Answer: r2cos 2 = 4
19
Convert to a rectangular equation.
76) r sin = 10
Answer: y = 10
77) r = 5
Answer: x2 + y2 = 25
Express the complex number in trigonometric form.
78) 6 2 - 6 2i
Answer: 12(cos 315° + i sin 315°)
Express in standard notation.
79) 8(cos 30° + i sin 30°)
Answer: 4 3 + 4i
Graph the equation.
80) r = 4 sin
Answer:
20