3 π 2 3 π 2
7 π 10 7 π 10
α = 150 ° α = 150 °
β = 60 ° β = 60 °
7 π 6 7 π 6
215 π 18 = 37.525 units 215 π 18 = 37.525 units
− 3 π 2 − 3 π 2 rad/s
1655 kilometers per hour
5.2 Unit Circle: Sine and Cosine Functions
cos ( t ) = − 2 2 , sin ( t ) = 2 2 cos ( t ) = − 2 2 , sin ( t ) = 2 2
cos ( π ) = − 1 , cos ( π ) = − 1 , sin ( π ) = 0 sin ( π ) = 0
sin ( t ) = − 7 25 sin ( t ) = − 7 25
approximately 0.866025403
- ⓐ cos ( 315 ° ) = 2 2 , sin ( 315 ° ) = – 2 2 cos ( 315 ° ) = 2 2 , sin ( 315 ° ) = – 2 2
- ⓑ cos ( − π 6 ) = 3 2 , sin ( − π 6 ) = − 1 2 cos ( − π 6 ) = 3 2 , sin ( − π 6 ) = − 1 2
( 1 2 , − 3 2 ) ( 1 2 , − 3 2 )
5.3 The Other Trigonometric Functions
sin t = − 2 2 sin t = − 2 2 , cos t = 2 2 cos t = 2 2 , tan t = − 1 tan t = − 1 , sec t = 2 sec t = 2 , csc t = − 2 csc t = − 2 , cot t = − 1 cot t = − 1
sin π 3 = 3 2 sin π 3 = 3 2 , cos π 3 = 1 2 cos π 3 = 1 2 , tan π 3 = 3 tan π 3 = 3 , sec π 3 = 2 sec π 3 = 2 , csc π 3 = 2 3 3 csc π 3 = 2 3 3 , cot π 3 = 3 3 cot π 3 = 3 3
sin ( − 7 π 4 ) = 2 2 , cos ( − 7 π 4 ) = 2 2 , tan ( − 7 π 4 ) = 1 , sin ( − 7 π 4 ) = 2 2 , cos ( − 7 π 4 ) = 2 2 , tan ( − 7 π 4 ) = 1 , sec ( − 7 π 4 ) = 2 , csc ( − 7 π 4 ) = 2 , cot ( − 7 π 4 ) = 1 sec ( − 7 π 4 ) = 2 , csc ( − 7 π 4 ) = 2 , cot ( − 7 π 4 ) = 1
sin t sin t
cos t = − 8 17 , sin t = 15 17 , tan t = − 15 8 cos t = − 8 17 , sin t = 15 17 , tan t = − 15 8 csc t = 17 15 , cot t = − 8 15 csc t = 17 15 , cot t = − 8 15
sin t = − 1 , cos t = 0 , tan t = Undefined sec t = Undefined, csc t = − 1 , cot t = 0 sin t = − 1 , cos t = 0 , tan t = Undefined sec t = Undefined, csc t = − 1 , cot t = 0
sec t = 2 , csc t = 2 , tan t = 1 , cot t = 1 sec t = 2 , csc t = 2 , tan t = 1 , cot t = 1
≈ − 2.414 ≈ − 2.414
5.4 Right Triangle Trigonometry
s i n t = 33 65 , cos t = 56 65 , t a n t = 33 56 , sec t = 65 56 , csc t = 65 33 , cot t = 56 33 s i n t = 33 65 , cos t = 56 65 , t a n t = 33 56 , sec t = 65 56 , csc t = 65 33 , cot t = 56 33
sin ( π 4 ) = 1 2 , cos ( π 4 ) = 1 2 , tan ( π 4 ) = 1 , sin ( π 4 ) = 1 2 , cos ( π 4 ) = 1 2 , tan ( π 4 ) = 1 , sec ( π 4 ) = 2 , c s c ( π 4 ) = 2 , cot ( π 4 ) = 1 sec ( π 4 ) = 2 , c s c ( π 4 ) = 2 , cot ( π 4 ) = 1
adjacent = 10 ; adjacent = 10 ; opposite = 10 3 opposite = 10 3 ; missing angle is π 6 π 6
About 52 ft
5.1 Section Exercises
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.
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.
4 π 3 4 π 3
2 π 3 2 π 3
7 π 2 ≈ 11.00 in 2 7 π 2 ≈ 11.00 in 2
81 π 20 ≈ 12.72 cm 2 81 π 20 ≈ 12.72 cm 2
π 2 π 2 radians
− 3 π − 3 π radians
π π radians
5 π 6 5 π 6 radians
5.02 π 3 ≈ 5.26 5.02 π 3 ≈ 5.26 miles
25 π 9 ≈ 8.73 25 π 9 ≈ 8.73 centimeters
21 π 10 ≈ 6.60 21 π 10 ≈ 6.60 meters
104.7198 cm 2
0.7697 in 2
8 π 9 8 π 9
1320 rad 210.085 RPM
7 in./s, 4.77 RPM, 28.65 deg/s
1 , 809 , 557.37 mm/min = 30.16 m/s 1 , 809 , 557.37 mm/min = 30.16 m/s
5.76 5.76 miles
120 ° 120 °
794 miles per hour
2,234 miles per hour
11.5 inches
5.2 Section Exercises
The unit circle is a circle of radius 1 centered at the origin.
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 t t and the horizontal axis.
The sine values are equal.
60° , 60° , Quadrant IV, sin ( 300° ) = − 3 2 , cos ( 300° ) = 1 2 sin ( 300° ) = − 3 2 , cos ( 300° ) = 1 2
45° , 45° , Quadrant II, sin ( 135° ) = 2 2 , sin ( 135° ) = 2 2 , cos ( 135° ) = − 2 2 cos ( 135° ) = − 2 2
60° , 60° , Quadrant II, sin ( 120° ) = 3 2 , sin ( 120° ) = 3 2 , cos ( 120° ) = − 1 2 cos ( 120° ) = − 1 2
30° , 30° , Quadrant II, sin ( 150° ) = 1 2 , sin ( 150° ) = 1 2 , cos ( 150° ) = − 3 2 cos ( 150° ) = − 3 2
π 6 , π 6 , Quadrant III, sin ( 7 π 6 ) = − 1 2 , sin ( 7 π 6 ) = − 1 2 , cos ( 7 π 6 ) = − 3 2 cos ( 7 π 6 ) = − 3 2
π 4 , π 4 , Quadrant II, sin ( 3 π 4 ) = 2 2 , sin ( 3 π 4 ) = 2 2 , cos ( 4 π 3 ) = − 2 2 cos ( 4 π 3 ) = − 2 2
π 3 , π 3 , Quadrant II, sin ( 2 π 3 ) = 3 2 , sin ( 2 π 3 ) = 3 2 , cos ( 2 π 3 ) = − 1 2 cos ( 2 π 3 ) = − 1 2
π 4 , π 4 , Quadrant IV, sin ( 7 π 4 ) = − 2 2 , sin ( 7 π 4 ) = − 2 2 , cos ( 7 π 4 ) = 2 2 cos ( 7 π 4 ) = 2 2
− 15 4 − 15 4
( − 10 , 10 3 ) ( − 10 , 10 3 )
( – 2.778 , 15.757 ) ( – 2.778 , 15.757 )
[ – 1 , 1 ] [ – 1 , 1 ]
sin t = 1 2 , cos t = − 3 2 sin t = 1 2 , cos t = − 3 2
sin t = − 2 2 , cos t = − 2 2 sin t = − 2 2 , cos t = − 2 2
sin t = 3 2 , cos t = − 1 2 sin t = 3 2 , cos t = − 1 2
sin t = − 2 2 , cos t = 2 2 sin t = − 2 2 , cos t = 2 2
sin t = 0 , cos t = − 1 sin t = 0 , cos t = − 1
sin t = − 0.596 , cos t = 0.803 sin t = − 0.596 , cos t = 0.803
sin t = 1 2 , cos t = 3 2 sin t = 1 2 , cos t = 3 2
sin t = − 1 2 , cos t = 3 2 sin t = − 1 2 , cos t = 3 2
sin t = 0.761 , cos t = − 0.649 sin t = 0.761 , cos t = − 0.649
sin t = 1 , cos t = 0 sin t = 1 , cos t = 0
− 6 4 − 6 4
( 0 , – 1 ) ( 0 , – 1 )
37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds
5.3 Section Exercises
Yes, when the reference angle is π 4 π 4 and the terminal side of the angle is in quadrants I and III. Thus, at x = π 4 , 5 π 4 , x = π 4 , 5 π 4 , the sine and cosine values are equal.
Substitute the sine of the angle in for y y in the Pythagorean Theorem x 2 + y 2 = 1. x 2 + y 2 = 1. Solve for x x and take the negative solution.
The outputs of tangent and cotangent will repeat every π π units.
2 3 3 2 3 3
− 2 3 3 − 2 3 3
− 3 3 − 3 3
If sin t = − 2 2 3 sin t = − 2 2 3 , sec t = − 3 sec t = − 3 , csc t = − 3 2 4 csc t = − 3 2 4 , tan t = 2 2 tan t = 2 2 , cot t = 2 4 cot t = 2 4
sec t = 2 sec t = 2 , csc t = 2 3 3 csc t = 2 3 3 , tan t = 3 tan t = 3 , cot t = 3 3 cot t = 3 3
− 2 2 − 2 2
sin t = 2 2 , cos t = 2 2 , tan t = 1 , cot t = 1 , sec t = 2 , csc t = 2 sin t = 2 2 , cos t = 2 2 , tan t = 1 , cot t = 1 , sec t = 2 , csc t = 2
sin t = − 3 2 sin t = − 3 2 , cos t = − 1 2 cos t = − 1 2 , tan t = 3 , cot t = 3 3 tan t = 3 , cot t = 3 3 , sec t = − 2 sec t = − 2 , csc t = − 2 3 3 csc t = − 2 3 3
sin ( t ) ≈ 0.79 sin ( t ) ≈ 0.79
csc t ≈ 1.16 csc t ≈ 1.16
sin t cos t = tan t sin t cos t = tan t
13.77 hours, period: 1000 π 1000 π
7.73 inches
5.4 Section Exercises
The tangent of an angle is the ratio of the opposite side to the adjacent side.
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.
b = 20 3 3 , c = 40 3 3 b = 20 3 3 , c = 40 3 3
a = 10 , 000 , c = 10 , 000.5 a = 10 , 000 , c = 10 , 000.5
b = 5 3 3 , c = 10 3 3 b = 5 3 3 , c = 10 3 3
5 29 29 5 29 29
5 41 41 5 41 41
c = 14 , b = 7 3 c = 14 , b = 7 3
a = 15 , b = 15 a = 15 , b = 15
b = 9.9970 , c = 12.2041 b = 9.9970 , c = 12.2041
a = 2.0838 , b = 11.8177 a = 2.0838 , b = 11.8177
a = 55.9808 , c = 57.9555 a = 55.9808 , c = 57.9555
a = 46.6790 , b = 17.9184 a = 46.6790 , b = 17.9184
a = 16.4662 , c = 16.8341 a = 16.4662 , c = 16.8341
498.3471 ft
368.7633 ft
Review Exercises
− 7 π 6 − 7 π 6
10.385 meters
2 π 11 2 π 11
1036.73 miles per hour
2 2 2 2 or − 2 2 − 2 2
sine, cosecant, tangent, cotangent
b = 8 , c = 10 b = 8 , c = 10
11 157 157 11 157 157
a = 4 , b = 4 a = 4 , b = 4
Practice Test
6.283 centimeters
3.351 feet per second, 2 π 75 2 π 75 radians per second
− 3 2 − 3 2
a = 9 2 , b = 9 3 2 a = 9 2 , b = 9 3 2
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Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
- Authors: Jay Abramson
- Publisher/website: OpenStax
- Book title: Precalculus
- Publication date: Oct 23, 2014
- Location: Houston, Texas
- Book URL: https://openstax.org/books/precalculus/pages/1-introduction-to-functions
- Section URL: https://openstax.org/books/precalculus/pages/chapter-5
© Dec 8, 2021 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.
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