Copyright © Cengage Learning. All rights reserved. 6 Trigonometry Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. 6.1 ANGLES AND THEIR MEASURE Copyright © Cengage Learning. All rights reserved.

What You Should Learn Use degree measure. Use radian measure. Convert between degree and radian measures.

Degree Measure

Degree Measure Quadrants: Type of angle: Right angle: quarter revolution Acute angle: between 0 and 90 Obtuse angle: between 90 and 180 Full revolution Straight angle: half revolution

Degree Measure Two positive angles  and  are complementary (complements of each other) if their sum is 90. Two positive angles are supplementary (supplements of each other) if their sum is 180. See Figure 6.11. Complementary angles Supplementary angles

Radian Measure

Arc length,s = radius, r when  = 1 radian Radian Measure Arc length,s = radius, r when  = 1 radian

Conversion of Angle Measure

Conversion of Angle Measure From the latter equation, you obtain and Which lead to the conversion rules.

Conversion of Angle Measure When no units of angle measure are specified, radian measure is implied. For instance, if you write  = 2, you imply that  = 2 radians. Figure 6.17

Example 4 – Converting from Degrees to Radians b. c. Multiply by  / 180. Multiply by  / 180. Multiply by  / 180.

Exercise Exercise 63

Example 5: Converting from Radians to Degrees Page 447 Exercise 67

Applications

Applications The radian measure formula,  = s / r, can be used to measure arc length along a circle.

Example 6 Page 449 Exercise 93

Applications

Example 9 – Area of a Sector of a Circle A sprinkler on a golf course fairway sprays water over a distance of 70 feet and rotates through an angle of 120 (see Figure 6.22). Find the area of the fairway watered by the sprinkler. Figure 6.22

Example 9 – Solution First convert 120 to radian measure as follows.  = 120 Then, using  = 2 /3 and r = 70, the area is Multiply by  /180. Formula for the area of a sector of a circle.

Example 9 – Solution cont’d Substitute for r and . Simplify.

Exercise Exercise 115

What You Should Learn Evaluate trigonometric functions of acute angles. Use fundamental trigonometric identities. Use a calculator to evaluate trigonometric functions.

The Six Trigonometric Functions

The Six Trigonometric Functions From a right triangle : (page 456)

Example 1 – Evaluating Trigonometric Functions Use the triangle in Figure 6.24 to find the values of the six trigonometric functions of . Solution: By the Pythagorean Theorem, (hyp)2 = (opp)2 + (adj)2, it follows that Figure 6.24

Example 1 – Solution So, the six trigonometric functions of  are cont’d So, the six trigonometric functions of  are

Exercise Use the Pythagorean Theorem to find the third side of the triangle. Hence, find the values of the six trigonometric functions of the angle   1. 3.             2. 4.            

Example 2 – Evaluating Trigonometric Functions of 45˚         Figure 6.24    

Example 3 – Evaluating Trigonometric Functions of 30˚ and 60˚.            

The Six Trigonometric Functions

The Six Trigonometric Functions If  is an acute angle, the following relationships are true. sin(90 –  ) = cos  cos(90 –  ) = sin  tan(90 –  ) = cot  cot(90 –  ) = tan  sec(90 –  ) = csc  csc(90 –  ) = sec 

Trigonometric Identities

Trigonometric Identities Note: sin2 =(sin)2 cos2 =(cos  )2

Example 4 – Applying Trigonometric Identities Let  be an acute angle such that sin  = 0.6. Find the values of cos  tan  using trigonometric identities. Solution: a. use the Pythagorean identity sin2  + cos2  = 1. (0.6)2 + cos2  = 1 cos2  = 1 – (0.6)2 = 0.64 Substitute 0.6 for sin . Subtract (0.6)2 from each side.

Example 4 – Solution cos  = = 0.8. b. = 0.75. cont’d Extract the positive square root. (Given in question  an acute angle) = 0.75.

Exercise  

Example 5: Applying Trigonometric Identities  

Exercise  

Evaluating Trigonometric Functions with a Calculator

Evaluating Trigonometric Functions with a Calculator To evaluate csc( /8), use the fact that and enter the following keystroke sequence in radian mode. Display 2.6131259

Example 6 – Using a Calculator Function Mode Calculator Keystrokes Display a. sin 76.4 Degree 0.9719610 b. cot 1.5 Radian 0.0709148 or Display 0.0709148

What You Should Learn Find reference angles. Evaluate trigonometric functions of real numbers.

Reference Angles

Reference Angles The values of the trigonometric functions of angles greater than 90 (or less than 0) can be determined from their values at corresponding acute angles called reference angles. .  ′ =  –   ′ = 180 –   ′ =  –   ′ =  – 180  ′ = 2 –   ′ = 360 – 

Example 4 – Finding Reference Angles Find the reference angle  ′. a.  = 300 b.  = 2.3 c.  = –135

Example 4(a) – Solution a)  ′ = 360 – 300 = 60 b)  ′ =  – 2.3  0.8416. a)  ′ = 360 – 300 = 60

Example 4(c) – Solution cont’d  ′ = 225 – 180 = 45.

Exercise