final exam review · 1, 7p 4) 0 p 6 p 3 p 2p2 3 5p 6 p 7p 6 4p 3p 2 5p 3 11p 6 1234 142) (2, 225°)...

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Worksheet by Kuta Software LLC

Precalculus

Final Exam Review

Name___________________________________ ID: 1

Date________________ Period____©c X2p0w1o8c FK_uRtLaZ fS_oPfYtDwlavrAeI WLFLCC[.e y UAQl[lE ar_iWgCh]t^sH tr[eeshemruvyendh.

-1-

Use identities to find the value of each expression.

1) If sin (q - p

2 ) = 0.52, find cos q . 2) If sin (-q ) = 0.16, find cos (q - p

2 ).

3) Find csc q and tan q

if cot q = 9

5 and sec q > 0.

4) Find tan q and sec q

if sin q = 3

4 and cot q > 0.

Verify each identity.

5) sec2 x + cot2 x = tan2 x + csc2 x 6)

csc2 x

tan2 x =

cot2 x

sin2 x

7) sec x + cos x = cos2 x + 1

cos x

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-2-

8) cot2 xsin x

= csc x

tan2 x 9)

cot x

sec2 x =

cos3 xsin x

10) tan x

sec3 x = cos2 xsin x

11) 2cos2 x(1 - cos 2x) = sin2 2x

12) 2sin2 xcos x = sin 2xcsc x

13) cos xtan 2x

= cos 2x2sin x

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-3-

14) tan x + sec2 x = 2(sin xcos x + 1)

1 + cos 2x

Use the half-angle identities to find the exact value of each.

15) cos 22.5° 16) sin 67.5°

Find the exact value of each.

17) tan q = -4

3 where

7p

2 £ q < 4p

Find sinq

2

18) cos q = 4

5 where 0 £ q <

p

2

Find sinq

2

Find the exact value of each expression.

19) tan-1 020) tan-1 3

3

21) tan-1 322) tan-1 -

3

3

23) cos-1 0 24) cos-1 1

2

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-4-

25) tan-1 126) cos-1 2

2

27) sin-1 -3

2

28) cos-1 1

29) cos-1 3

2

30) sin-1 1

31) cot sin-1 3 34

3432) tan-1 (cos

p

2 )

33) csc sin-1 57

1134) sin-1 (cot

3p

4 )

35) cos-1 (cotp

4 ) 36) tan sin-1 6 2

11

37) cos-1 (sec 0)38) sin cos-1 3 11

11

39) sin-1 (tan 0)40) csc sin-1 13

7

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-5-

41) tan-1 (sec p)42) sec sin-1 4

5

Write each trigonometric expression as an algebraic expression.

43) cot cos-1 x 44) csc sin-1 x

45) sin cos-1 x 46) csc tan-1 x

47) cot sin-1 x 48) sec sin-1 x

Find the exact value of each.

49) sin2p

9cos

p

18 - cos

2p

9sin

p

1850)

tan43p

18 - tan

5p

9

1 + tan43p

18tan

5p

9

51) sin 159cos 24 - cos 159sin 24 52) sin 104cos 46 + cos 104sin 46

Simplify.

53) cos 2q cos -q + sin 2q sin -q54)

tan 5q + tan -6q

1 - tan 5q tan -6q

55) cos 5vcos 6v - sin 5vsin 6v56)

tan -2x - tan 6x1 + tan -2xtan 6x

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-6-

Write each trigonometric expression as an algebraic expression.

57) cos (tan-1 x + tan-1 0)58) tan (tan-1 x - sin-1 2

2 )

Verify each identity.

59) sin (q - 3p

2 ) = cos q 60) sin (p

2 + q ) = cos q

State the number of possible triangles that can be formed using the given measurements.

61) In ZXY, Zm = 63°, y = 30, z = 6 62) In TRS, Tm = 138°, s = 26, t = 20

63) In HPK, Hm = 66°, k = 28, h = 8 64) In FDE, Em = 55°, e = 21, d = 4

Find each measurement indicated. Round your answers to the nearest tenth.

65) In DEF, Dm = 151°, Em = 9°, d = 34Find e

66) In BCA, Bm = 57°, a = 22, b = 5Find c

67) In DEF, Dm = 31°, f = 27, d = 20Find e

68) In STR, Tm = 36°, Rm = 30°, s = 42Find r

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-7-

69) In KHP, Km = 139°, p = 23, k = 27Find Pm

70) In ZXY, Ym = 89°, y = 16, x = 8Find Xm

71) In ABC, Cm = 73°, c = 20, b = 14Find Bm

72) In BCA, Bm = 16°, a = 29, b = 8Find Cm

Solve each triangle. Round your answers to the nearest tenth.

73) In YZX, Ym = 75°, x = 21, y = 8 74) In STR, Sm = 137°, Rm = 23°, r = 8

75)

30 m

13 m

AB

C

73°

76)

29 mi

13 mi

21 miB

C

A

77)

28.7 in29.5 in

B

CA

89°

78)

24 in

21 in

CA

B

89°

79) a = 24 mi, b = 15 mi, Cm = 107° 80) a = 21 in, b = 23 in, c = 8 in

81) a = 28 km, c = 18 km, b = 12 km 82) b = 24.5 m, c = 25.8 m, a = 23.4 m

Find each measurement indicated. Round your answers to the nearest tenth.

83) c = 16.6 in, b = 20.1 in, Am = 119.1°Find a

84) Cm = 128°, a = 28 mi, b = 19 miFind c

85) c = 10 yd, b = 17 yd, Am = 25°Find a

86) b = 15 in, Am = 31°, c = 28 inFind a

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-8-

Find the area of each triangle to the nearest tenth.

87)

9 in AB

C

112°22°

88)

10 m TR

S

84°65°

89)

5 cm RS

T

137°21°

90) 10 m

A

BC

59°

58°

91)

11 yd

15 yd

B

CA

34°

92)

4 in 5.3 in

T

R

S39°

93)

6 yd

9 yd

B

C

A127°

94)

5 km

8 km

F

DE

62°

Find the component form of the resultant vector.

95) u = 2, 4g = 0, 4Find: u - g

96) u = -4, -10Find the vector opposite u

97) f = -2, -5Find: -4f

98) Given: T = (9, 0) X = (-1, -4) Y = (-1, 0) Z = (-7, 4)Find: TX + 4YZ

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-9-

99) Given: A = (5, -5) B = (7, 3)Find: 7AB

100) Given: A = (-3, 0) B = (-9, -10)Find the vector opposite AB

Express the resultant vector as a linear combination of unit vectors i and j.

101) Given: A = (-5, -7) B = (0, 4) C = (3, 8) D = (8, -6)Find: 7AB - 9CD

102) u = -5i + 7jFind the vector opposite u

103) f = -8i + 2jFind the vector opposite f

104) f = -8i - 8jUnit vector in the opposite direction of f

105) f = 30i - 1309 jFind the vector opposite f

106) Given: A = (9, 7) B = (2, -10)Find: 3 × AB

107) u = -6i + 2jUnit vector in the direction of u

Find the component form of the resultant vector.

108) u = 1, 3 2Unit vector in the direction of u

Find the magnitude and direction angle of the resultant vector.

109) f = -4, -12Unit vector in the direction of f

110) u = -9, 40Unit vector in the direction of u

111) u = -12, 11Unit vector in the opposite direction of u

112) a = 15, 36Unit vector in the opposite direction of a

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-10-

113) f = -2, 7g = 10, -6Find: -7f - 7g

114) f = -9, -8b = 0, -11Find: -2f - 5b

115) u = -5, -4Unit vector in the direction of u

116) f = 12, -10b = -2, 6Find: -10f + 6b

Find the component form, magnitude, and direction angle of the resultant vector.

117) f = 12, 7g = 5, 3Find: -6f + 2g

118) a = 12, -4g = 12, 12Find: 4a + 8g

Express the resultant vector as a linear combination of unit vectors i and j, and find themagnitude and direction angle.

119) f = 4ig = 8i - jFind: -8f + 4g

120) u = -8i + 11jv = -i - 12jFind: -10u + 5v

Draw a diagram to illustrate the horizontal and vertical components of the vector. Then find themagnitude of each component.

121) a = 11, 280° 122) m = 29, 37°

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-11-

Draw a vector diagram to find the resultant of each pair of vectors using the triangle method. Then state the magnitude and direction angle of the resultant.

123) t = -5, 12 u = 8, 15 124) t = 8, 15 u = 5, -12

Draw a vector diagram to find the resultant of each set of vectors. Then state the magnitudeand direction angle of the resultant.

125) a = -14, 12 b = 5, 13 c = 5, -12 126) t = 5, -12 u = -8, 6 v = -8, 15

Use the given vectors to draw a vector diagram for each expression using the triangle method. Then state the magnitude and direction angle of the resultant.

127) a = -8, 15 b = 19, 29a + b

128) m = 12, 16 n = 1, -10-8m - 3n

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-12-

Use the given vectors to draw a vector diagram for each expression using the parallelogrammethod. Then state the magnitude and direction angle of the resultant.

129) a = -10, 1 b = 6, -5-7a - 9b

130) a = 12, 16 b = 3, -142a + 3b

Find the dot product of the given vectors.

131) u = 2, -4v = -3, 5

132) u = 9, 2v = -5, -7

133) u = 2i + 9jv = -7i + j

134) u = 3i - 4jv = -3i - 9j

State if the two vectors are parallel, orthogonal, or neither.

135) u = 7i + 2jv = 2i + 7j

136) u = -3i + 3jv = 16i - 12j

Find the measure of the angle between the two vectors.

137) u = 8i + 5jv = 6i + 8j

138) u = -9i - 7jv = -8i + 7j

139) u = -8i - 3jv = -8i + 5j

140) u = 6i - 5jv = i + j

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-13-

Find all pairs of polar coordinates that describe the same point as the provided polarcoordinates.

141) (1, 7p

4 )

0

p

6

p

3

p

22p

3

5p

6

p

7p

6

4p

3 3p

2

5p

3

11p

6

1 2 3 4

142) (2, 225°)

0°

30°

60°90°

120°

150°

180°

210°

240°

270°

300°

330°

1 2 3 4

Convert each pair of polar coordinates to rectangular coordinates.

143) (3, 300°)144) (3,

7p

4 )

Convert each pair of rectangular coordinates to polar coordinates where r > > 00 and 00 £ £ qq < < 22pp .

145) ( 2

2, -

2

2 ) 146) (1

2,

3

2 )

Two points are specified using polar coordinates. Find the distance between the points.

147) (2, 11p

6 ), (2, 7p

12 ) 148) (1, 3p

2 ), (3, p

4 )

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-14-

Convert each equation from polar to rectangular form. Then graph the polar equation.

149) tan q = 5

0

p

6

p

3

p

22p

3

5p

6

p

7p

6

4p

3 3p

2

5p

3

11p

6

1 2 3 4 5 6 7

150) cot q = 5

0

p

6

p

3

p

22p

3

5p

6

p

7p

6

4p

3 3p

2

5p

3

11p

6

1 2 3 4 5 6 7

151) tan q = 4

0

p

6

p

3

p

22p

3

5p

6

p

7p

6

4p

3 3p

2

5p

3

11p

6

1 2 3 4 5 6 7

152) q = p

6

0

p

6

p

3

p

22p

3

5p

6

p

7p

6

4p

3 3p

2

5p

3

11p

6

1 2 3 4 5 6 7

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-15-

Consider each polar equation over the given interval. Classify the curve; determine if the graphis symmetric with respect to the origin, polar axis, and line qq =pp /22; find the values of qq where r is zero; find the maximum r value and the values of qq where this occurs; and sketch thegraph.

153) r2 = 25sin (2q ), 0 ≤ q < 2p

0

p

6

p

3

p

22p

3

5p

6

p

7p

6

4p

3 3p

2

5p

3

11p

6

1 2 3 4 5 6 7

154) r2 = 4sin (2q ), 0 ≤ q < 2p

0

p

6

p

3

p

22p

3

5p

6

p

7p

6

4p

3 3p

2

5p

3

11p

6

1 2 3 4 5 6 7

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-16-

155) r2 = 16cos (2q ), 0 ≤ q < 2p

0

p

6

p

3

p

22p

3

5p

6

p

7p

6

4p

3 3p

2

5p

3

11p

6

1 2 3 4 5 6 7

156) r2 = 49sin (2q ), 0 ≤ q < 2p

0

p

6

p

3

p

22p

3

5p

6

p

7p

6

4p

3 3p

2

5p

3

11p

6

1 2 3 4 5 6 7

Convert numbers in rectangular form to polar form and numbers in polar form to rectangularform.

157) 4(cosp

4 + isin

p

4 ) 158) 3(cos 45 + isin 45)

Simplify. Write your answer in rectangular form when rectangular form is given and in polarform when polar form is given.

159) 4(cos 210 + isin 210) × 5(cos 120 + isin 120)

160) (6 + 6i)(-2 + 4i)161)

6(cos 60 + isin 60)3(cos 225 + isin 225)

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-17-

162) 15(cos 300 + isin 300)

2 5(cos 240 + isin 240)163) (5(cos 60 + isin 60))3

164) (2(cos 60 + isin 60))4

Find the absolute value.

165) 6(cos 90 + isin 90) 166) 2 2 - 2i 2

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-18-

Answers to Final Exam Review (ID: 1)

1) -0.52 2) -0.163)

106

5 and

5

94)

3 7

7 and

4 7

7

5) sec2 x + cot2 x Use tan2 x + 1 = sec2 x

tan2 x + 1 + cot2 x Use cot2 x + 1 = csc2 x

tan2 x + csc2 x ■

6) csc2 x

tan2 xUse cot x =

1

tan x

csc2 xcot2 x Use csc x = 1

sin x

cot2 x

sin2 x■

7) sec x + cos x Decompose into sine and cosine

1

cos x + cos x Simplify

cos2 x + 1

cos x■

8) cot2 xsin x

Use cot x = 1

tan x

1

tan2 xsin xUse csc x =

1

sin x

csc x

tan2 x■

9) cot x

sec2 xUse cot x =

cos xsin x

cos x

sec2 xsin xUse sec x =

1

cos x

cos3 xsin x

■

10) tan x

sec3 xDecompose into sine and cosine

sin xcos x

( 1

cos x)3Simplify

cos2 xsin x ■

11) 2cos2 x(1 - cos 2x) Use cos 2x = 1 - 2sin2 x

4cos2 xsin2 x Use sin 2x = 2sin xcos x

sin2 2x ■

12) 2sin2 xcos x Use sin 2x = 2sin xcos x

sin xsin 2x Use csc x = 1

sin x

sin 2xcsc x

■

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-19-

13) cos xtan 2x

Use tan 2x = sin 2xcos 2x

cos xcos 2xsin 2x

Use sin 2x = 2sin xcos x

cos xcos 2x2sin xcos x

Cancel common factors

cos 2x2sin x

■

14) tan x + sec2 x Decompose into sine and cosine

sin xcos x

+ ( 1

cos x)2

Simplify

sin xcos x + 1

cos2 xUse cos2 x =

1 + cos 2x2

2(sin xcos x + 1)1 + cos 2x

■

15) 2 + 2

216)

2 + 2

217) -

5

518)

10

10

19) 020)

p

621)

p

322) -

p

6

23) p

224)

p

325)

p

426)

p

4

27) -p

3

28) 029)

p

630)

p

2

31) 5

3

32) 033)

11 57

5734) -

p

2

35) 036)

6 2

7

37) 038)

22

1139) 0

40) 7 13

1341) -

p

442)

5

3

43) x

1 - x244)

1

x45) 1 - x2

46) 1 + x2

x

47) 1 - x2

x48)

1

1 - x249)

1

250) -

3

3

51) 2

252)

1

2

53) cos 3q 54) tan -q

55) cos 11v 56) tan -8x57)

x2 + 1

x2 + 158)

x - 1

1 + x

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-20-

59) sin (q - 3p

2 )= sin q cos

3p

2 - cos q sin

3p

2= sin q × 0 - cos q × -1= cos q

60) sin (p

2 + q )

= sinp

2cos q + cos

p

2sin q

= cos q + 0sin q= cos q

61) None

62) None 63) None 64) One triangle 65) 1166) Not a triangle 67) 37.5 or 8.8 68) 23 69) 34°70) 30° 71) 42° 72) 76.3° or 71.7° 73) Not a triangle74) Tm = 20°, s = 14, t = 7 75) Bm = 25.4°, Cm = 81.6°, a = 29 m76) Bm = 24°, Cm = 41°, Am = 115° 77) Cm = 46.3°, Am = 44.7°, b = 40.8 in78) Am = 41.6°, Bm = 49.4°, c = 31.6 in 79) Am = 46.2°, Bm = 26.8°, c = 31.8 mi80) Am = 65.6°, Bm = 94.1°, Cm = 20.3° 81) Cm = 26°, Am = 137°, Bm = 17°82) Bm = 59.5°, Cm = 65.1°, Am = 55.4° 83) 31.7 in 84) 42.4 mi85) 9 yd 86) 17 in 87) 19.6 in² 88) 87.5 m²89) 8.2 cm² 90) 44.1 m² 91) 70.1 yd² 92) 9.8 in²93) 9.6 yd² 94) 19.9 km² 95) 2, 0 96) 4, 1097) 8, 20 98) -34, 12 99) 14, 56 100) 6, 10101) -10i + 203j 102) 5i - 7j 103) 8i - 2j

104) 2 × i2

+ 2 × j2

105) -30i + 1309 × j 106) -7 3 × i - 17 3 × j107) -

3 10 × i10

+ 10 × j10

108) 19

19,

3 38

19

109) 251.57° 110) 102.68° 111) 317.49°

112) 247.38° 113) 7 65 » 56.436; 187.13° 114) 5365 » 73.246; 75.77°115) 218.66° 116) 4 2245 » 189.526; 134.14°117) -62, -36

2 1285 » 71.694; 210.14°118) 144, 80

16 106 » 164.73; 29.05°119) -4j

4; 270°120) 75i - 170j

5 1381 » 185.809; 293.81°121)

a y

x

Horizontal: 1.91Vertical: -10.83

122)

my

x

Horizontal: 23.16Vertical: 17.45

123)

t

u

t + u

27.17; 83.66°

124)

tu

t + u

13.34; 12.99°

125)

a

b c

a + b + c

13.6; 107.1°

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-21-

126)

t

u

v

t + u + v

14.21; 140.71°

127)

9a

b

9a + b

146.89; 111.15°

128)

-8m

-3n

-8m - 3n

139.3; 224.71°

129)

-7a

-9b-7a - 9b

41.23; 67.17°

130)

2a

3b

2a + 3b

34.48; 343.14°

131) -26

132) -59 133) -5 134) 27 135) Neither136) Neither 137) 21.12° 138) 79.06° 139) 52.56°140) 84.81°

141) (1, 7p

4 + 2np) and (-1,

7p

4 + (2n + 1)p)

where n is an integer142) (2, 225° + 360n°) and (-2, 45° + 360n°)

where n is an integer 143) (3

2, -

3 3

2 ) 144) (3 2

2, -

3 2

2 )145) (1,

7p

4 ) 146) (1, p

3 ) 147) 2 2 + 2 » 3.696

148) 10 + 3 2 » 3.774 149)

0

p

6

p

3

p

22p

35p

6

p

7p

64p

33p

2

5p

3

11p

6

2 4 6

y = 5x150)

0

p

6

p

3

p

22p

35p

6

p

7p

64p

33p

2

5p

3

11p

6

2 4 6

y = x5

151)

0

p

6

p

3

p

22p

35p

6

p

7p

64p

33p

2

5p

3

11p

6

2 4 6

y = 4x152)

0

p

6

p

3

p

22p

35p

6

p

7p

64p

33p

2

5p

3

11p

6

2 4 6

y = x 3

3

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-22-

153)

0

p

6

p

3

p

22p

35p

6

p

7p

64p

33p

2

5p

3

11p

6

2 4 6

LemniscateSymmetric about the origin

r = 0 when q = {0, p

2, p,

3p

2 }r = 5 when q = {p

4,

5p

4 }

154)

0

p

6

p

3

p

22p

35p

6

p

7p

64p

33p

2

5p

3

11p

6

2 4 6


r = 0 when q = {0, p

2, p,

3p

2 }r = 2 when q = {p

4,

5p

4 }

155)

0

p

6

p

3

p

22p

35p

6

p

7p

64p

33p

2

5p

3

11p

6

2 4 6

LemniscateSymmetric about the origin,

polar axis, and line q = p

2

r = 0 when q = {p

4,

3p

4,

5p

4,

7p

4 }r = 4 when q = {0, p}

156)

0

p

6

p

3

p

22p

35p

6

p

7p

64p

33p

2

5p

3

11p

6

2 4 6


r = 0 when q = {0, p

2, p,

3p

2 }r = 7 when q = {p

4,

5p

4 }

157) 2 2 + 2i 2158)

3 2

2 +

3 2

2i

159) 20(cos 330 + isin 330)

160) -36 + 12i 161) 2(cos -165 + isin -165)162)

3

2(cos 60 + isin 60)

163) 125(cos 180 + isin 180) 164) 16(cos 240 + isin 240) 165) 6166) 4

final exam review · 1, 7p 4) 0 p 6 p 3 p 2p2 3 5p 6 p 7p 6 4p 3p 2 5p 3 11p 6 1234 142) (2, 225°)...

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