Unit 5 trigonometric functions homework 5 solving right triangles answer key

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5.1 Angles

8.

215π18=37.525 units 215π18=37.525 units

11.

1655 kilometers per hour

5.2 Unit Circle: Sine and Cosine Functions

1.

cos(t)=−2 2,sin(t)=22cos(t)=−22,sin(t)= 22

2.

cos(π)=−1,cos(π)=−1, sin(π)=0sin(π)=0

3.

sin(t)=−725 sin(t)=−725

4.

approximately 0.866025403

6.

1. ⓐcos(315°)= 22, sin(315°)=–22cos(315°)=22, sin (315°)=–22
2. ⓑcos(−π6)=32, sin( −π6)=−12cos(−π6)=32, sin(−π6)=− 12

7.

(12,−32)(12,−32)

5.3 The Other Trigonometric Functions

1.

sint=−22sint=−22 , cos t=22cost=22, tant=−1tant=−1, sect=2sect=2, csct=−2csct=−2, cott=−1cot t=−1

2.

sinπ3=32sinπ3=32, cosπ3=12 cosπ3=12, tanπ3=3tanπ3=3, secπ3=2secπ3=2, cscπ3=233 cscπ3=233, cotπ3=33cotπ3=33

3.

sin( −7π4)=22,cos(−7π4)=22,tan(−7π4 )=1,sin(−7π4)=22,cos(−7π4)=2 2,tan(−7π4)=1,
sec(−7π4)=2,csc (−7π4)=2,cot(−7π4)=1sec(−7π4) =2,csc(−7π4)=2,cot(−7π4)=1

7.

cost=−817,sin t=1517,tant=−158cost=−817,sint= 1517,tant=−158
csct=1715,cott=−815csct=1715,cott=−815

8.

sint= −1,cost=0,tant=Undefinedsect= Undefined,csct=−1,cott=0 sint=−1,cost=0,tant=Undefinedsect= Undefined,csc t=−1,cott=0

9.

sect=2,csct =2,tant=1,cott=1sect=2,csct=2,tan t=1,cott=1

5.4 Right Triangle Trigonometry

2.

sint= 3365,cost=5665,tant=3356,sect=6556,csct=6533,cott=5633sint=3365,cost=5665,tant=3356, sect=6556,csct=6533,cott=5633

3.

sin (π4)=12,cos(π4)=12,tan(π4)=1,sin(π4)=12,cos(π4)=12,tan(π4)=1 ,
sec(π4)=2,csc(π4)=2,cot(π4 )=1sec(π4)=2,csc(π4)=2,cot(π4) =1

5.

adjacent =10;adjacent=10; opposite=103opposite=103; missing angle is π6π6

5.1 Section Exercises

3.

Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

5.

Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

19.

4π3 4π3

21.

2π32π 3

23.

7π2≈11.00 in27π2≈11.00 in2

25.

81π20≈12.72 cm281π20≈12.72 cm2

41.

5.02π3≈5.265.02π3 ≈5.26 miles

43.

25π 9≈8.7325π9≈8.73 centimeters

45.

21π10≈6.60 21π10≈6.60 meters

61.

7 in./s, 4.77 RPM, 28.65 deg/s

63.

1,809,557.37 mm/min =30.16 m/s1,809,557.37 mm/min=30.16 m/s

5.2 Section Exercises

1.

The unit circle is a circle of radius 1 centered at the origin.

3.

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, t,t, formed by the terminal side of the angle tt and the horizontal axis.

5.

The sine values are equal.

35.

60°,60°, Quadrant IV, sin(300°)=−32,cos(300°)=12 sin(300°)=−32,cos(300°)=12

37.

45°,45°, Quadrant II, sin(135°)=22,sin(135°)=22, cos(135°)=−22cos(135°)=−22

39.

60°, 60°, Quadrant II, sin(120°)=32,sin(120°)=32, cos(120°)=−12cos(120°)=− 12

41.

30°,30°, Quadrant II, sin(150°)=12,sin(150°)=12, cos(150°)=−32cos(150°)=−32

43.

π6,π6 , Quadrant III, sin(7π6)=−12,sin(7π6)=−12, cos(7π6)=−32cos(7π6)=−32

45.

π4,π4 , Quadrant II, sin(3π4)=22,sin(3π4 )=22, cos(4π3)=−22cos (4π3)=−22

47.

π3,π3 , Quadrant II, sin(2π3)=32,sin(2π3 )=32, cos(2π3)=−12cos(2π3)=−12

49.

π4,π4 , Quadrant IV, sin(7π4)=−22,sin(7π 4)=−22, cos(7π4)=22 cos(7π4)=22

55.

(−10,103) (−10,103)

57.

(–2.778,15.757)(–2.778,15.757)

61.

sint=12,cost=− 32sint=12,cost=−32

63.

sint=−22 ,cost=−22sint=−22,cost=−22

65.

sin t=32,cost=−12sint=32,cost= −12

67.

sint=−22,cost=22sint=−22 ,cost=22

69.

sint=0,cost =−1sint=0,cost=−1

71.

sint=−0.596,cos t=0.803sint=−0.596,cost=0.803

73.

sint=12,cost=32sint=12,cost=32

75.

sint=−1 2,cost=32sint=−12,cost=32

77.

sin t=0.761,cost=−0.649sint=0.761,cost=−0.649

79.

sint=1,cost =0sint=1,cost=0

103.

37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

5.3 Section Exercises

1.

Yes, when the reference angle is π4 π4 and the terminal side of the angle is in quadrants I and III. Thus, at x=π4,5π4,x=π4 ,5π4, the sine and cosine values are equal.

3.

Substitute the sine of the angle in for yy in the Pythagorean Theorem x2+y2=1.x2+y2=1. Solve for x x and take the negative solution.

5.

The outputs of tangent and cotangent will repeat every ππ units.

39.

If sint=−223 sint=−223, sect=−3sect =−3, csct=−324csct=−324, tant=22tant=22, cott=24 cott=24

41.

sect=2sec t=2, csct=233csct=233 , tant=3tant=3, cott=3 3cott=33

49.

sint=22,cos t=22,tant=1,cott=1,sect=2,csct=2 sint=22,cost=22,tant=1,cott=1,sec t=2,csct=2

51.

sint=−32 sint=−32, cost=−12cost =−12, tant=3,cott=33tant=3 ,cott=33, sect=−2sect=−2, csct=−233csct=−233

63.

sin(t)≈0.79 sin(t)≈0.79

71.

sintcost=tantsintcost= tant

73.

13.77 hours, period: 1000π1000π

5.4 Section Exercises

3.

The tangent of an angle is the ratio of the opposite side to the adjacent side.

5.

For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

11.

b=2033,c =4033b=2033,c=4033

13.

a=10,000,c=10,000.5a=10,000,c=10,000.5

15.

b=533,c=10 33b=533,c=1033

29.

c=14,b=73 c=14,b=73

33.

b=9.9970,c=12.2041b=9.9970,c=12.2041

35.

a=2.0838,b=11.8177a=2.0838,b=11.8177

37.

a=55.9808,c=57.9555a =55.9808,c=57.9555

39.

a=46.6790,b=17.9184a=46.6790,b=17.9184

41.

a=16.4662,c=16.8341a =16.4662,c=16.8341

Review Exercises

15.

1036.73 miles per hour

35.

2 222 or −22−22

37.

sine, cosecant, tangent, cotangent

45.

11157157 11157157

Practice Test

9.

3.351 feet per second, 2π752π 75 radians per second

23.

a=92,b=932a=92,b=932

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