5 Trigonometric Functions Copyright Cengage Learning All rights

5 Trigonometric Functions Copyright © Cengage Learning. All rights reserved.

5. 1 Angles and Their Measure Copyright © Cengage Learning. All rights reserved.

What You Should Learn • Describe angles. • Use degree measure. • Use radian measure and convert between degrees and radians. • Use angles to model and solve real-life problems. 3

Angles 4

Angles As derived from the Greek language, the word trigonometry means “measurement of triangles. ” Initially, trigonometry dealt with relationships among the sides and angles of triangles and was used in the development of astronomy, navigation, and surveying. With the development of calculus and the physical sciences in the 17 th century, a different perspective arose—one that viewed the classic trigonometric relationships as functions having the set of real numbers as their domains. 5

Angles Consequently, the applications of trigonometry expanded to include a vast number of physical phenomena involving rotations and vibrations, including the following. • sound waves • light rays • planetary orbits • vibrating strings • pendulums • orbits of atomic particles 6

Angles This text incorporates both perspectives, starting with angles and their measure. An angle is determined by rotating a ray (half-line) about its endpoint. The starting position of the ray is the initial side of the angle, and the position after rotation is the terminal side, as shown in Figure 5. 1 7

Angles The endpoint of the ray is the vertex of the angle. This perception of an angle fits a coordinate system in which the origin is the vertex and the initial side coincides with the positive x-axis. Such an angle is in standard position, as shown in Figure 5. 2 8

Angles Positive angles are generated by counterclockwise rotation, and negative angles by clockwise rotation, as shown in Figure 5. 3 9

Angles are labeled with Greek letters such as (alpha), (beta), and (theta), as well as uppercase letters such as A, B and C. In Figure 5. 4, note that angles and have the same initial and terminal sides. Such angles are coterminal. Figure 5. 4 10

Degree Measure 11

Degree Measure The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. The most common unit of angle measure is the degree, denoted by the symbol . A measure of one degree (1 ) is equivalent to a rotation of of a complete revolution about the vertex. 12

Degree Measure To measure angles, it is convenient to mark degrees on the circumference of a circle, as shown in Figure 5. 5 13

Degree Measure So, a full revolution (counterclockwise) corresponds to 360 , a half revolution to 180 , a quarter revolution to 90 and so on. As we know that the four quadrants in a coordinate system are numbered I, III, and IV. 14

Degree Measure Figure 5. 6 shows which angles between 0 and 360 lie in each of the four quadrants. Figure 5. 6 15

Degree Measure Figure 5. 7 shows several common angles with their degree measures. Note that angles between 0 and 90 are acute and angles between 90 and 180 are obtuse. Figure 5. 7 16

Degree Measure Two angles are coterminal when they have the same initial and terminal sides. For instance, the angles 0 and 360 are coterminal, as are the angles 30 and 390. You can find an angle that is coterminal to a given angle by adding or subtracting 360 (one revolution), as demonstrated in Example 1. A given angle has infinitely many coterminal angles. For instance, = 30 is coterminal with 30 + n(360 ) where n is an integer. 17

Example 1 – Finding Coterminal Angles Find two coterminal angles (one positive and one negative) for (a) = 390 and (b) = – 120. Solution: a. For the positive angle = 390 , subtract 360 to obtain a positive coterminal angle. 390 – 360 = 30 See Figure 5. 8 18

Example 1 – Solution cont’d Subtract 2(360 ) = 720 to obtain a negative coterminal angle. 390 – 720 = – 330 b. For the negative angle = – 120 , add 360 to obtain a positive coterminal angle. – 120 + 360 = 240 See Figure 5. 9 19

Example 1 – Solution cont’d Subtract 360 to obtain a negative coterminal angle. – 120 – 360 = – 480 Two positive angles and are complementary (complements of each other) when their sum is 90. Two positive angles are supplementary (supplements of each other) when their sum is 180. (See Figure 5. 10. ) Figure 5. 10 20

Example 2 – Complementary and Supplementary Angles Find the complementary and supplementary angles of following angles. a. 72 b. 148 Solution: a. The complement of 72 is 90 – 72 = 18. The supplement of 72 is 180 – 72 = 180. 21

Example 2 – Solution cont’d b. Because 148 is greater than 90 , 148 has no complement. (Remember that complements are positive angles. ) The supplement is 180 – 148 = 32. 22

Radian Measure 23

Radian Measure A second way to measure angles is in radians. This type of measure is especially useful in calculus. To define a radian, you can use a central angle of a circle, one whose vertex is the center of the circle, as shown in Figure 5. 11 24

Radian Measure Because the circumference of a circle is 2 r units, it follows that a central angle of one full revolution (counterclockwise) corresponds to an arc length of s = 2 r. 25

Radian Measure Moreover, because 2 6. 28, there are just over six radius lengths in a full circle, as shown in Figure 5. 12 26

Radian Measure Because the units of measure for and are the same, the ratio s/r has no units—it is simply a real number. Because 2 radians corresponds to one complete revolution, degrees and radians are related by the equations 360 = 2 rad and 180 = rad. From the second equation, you obtain 27

Radian Measure This equation leads to the following conversion rules. Figure 5. 13 28

Radian Measure When no units of angle measure are specified, radian measure is implied. For instance, = or = 2 implies that = radians or = 2 radians. 29

Example 3 – Converting from Degrees to Radians a. Multiply by b. Multiply by 30

Example 4 – Converting from Radians to Degrees a. b. Multiply by 31

Linear and Angular Speed 32

Linear and Angular Speed The radian measure formula = s/r can be used to measure arc length along a circle. 33

Example 6 – Finding Arc Length A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240 , as shown in Figure 5. 17 34

Example 6 – Solution To use the formula s = r first convert 240 to radian measure. Convert from degrees to radians. Simplify. Then, using a radius of r = 4 inches, you can find the arc length to be s = r Length of circular arc 35

Example 6 – Solution cont’d Length of circular arc Simplify. 16. 76 inches. Use a calculator. Note that the units for r are determined by the units for r because is given in radian measure and therefore has no units. 36

Linear and Angular Speed The formula for the length of a circular arc can be used to analyze the motion of a particle moving at a constant speed along a circular path. 37

Example 8 – Finding Angular and Linear Speed A 15 -inch diameter tire on a car makes 9. 3 revolutions per second (see Figure 5. 19). Figure 5. 19 a. Find the angular speed of the tire in radians per second. b. Find the linear speed of the car. 38

Example 8 – Solution cont’d a. Because each revolution generates 2 radians, it follows that the tire turns (9. 3)(2 ) = 18. 6 radians per second. In other words, the angular speed is Angular speed = = 18. 6 radians per second. 39

Example 8 – Solution cont’d b. The linear speed of the tire and car is Linear speed = 438. 25 inches per second. 40