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Slide 1:

Copyright © Cengage Learning. All rights reserved. 4 Trigonometry

Slide 2:

4.4 TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Copyright © Cengage Learning. All rights reserved.

What You Should Learn:

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

Slide 4:

Introduction

Introduction:

Introduction The definitions of trigonometric functions were restricted to acute angles. In this section, the definitions are extended to cover any angle.

Introduction:

Introduction Because r = cannot be zero, it follows that the sine and cosine functions are defined for any real value of  . However, if x = 0, the tangent and secant of  are undefined. For example, the tangent of 90  is undefined. Similarly, if y = 0, the cotangent and cosecant of  are undefined.

Example 1 – Evaluating Trigonometric Functions:

Example 1 – Evaluating Trigonometric Functions Let (–3, 4) be a point on the terminal side of  . Find the sine, cosine, and tangent of  . Solution: Referring to Figure 4.36, you can see that x = –3, y = 4, and Figure 4.36

Example 1 – Solution:

Example 1 – Solution So, you have the following. cont’d

Introduction:

Introduction The signs of the trigonometric functions in the four quadrants can be determined from the definitions of the functions. For instance, because cos  = x / r , it follows that cos  is positive wherever x  0, which is in Quadrants I and IV. (Remember, r is always positive.)

Introduction:

Introduction In a similar manner, you can verify the results shown in Figure 4.37. Figure 4.37

Slide 11:

Reference Angles

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.

Reference Angles:

Reference Angles Figure 4.39 shows the reference angles for  in Quadrants II, III, and IV.  ′ =  –  (radians)  ′ = 180  –  (degrees)  ′ =  –  (radians)  ′ =  – 180  (degrees)  ′ = 2  –  (radians)  ′ = 360  –  (degrees) Figure 4.39

Example 4 – Finding Reference Angles:

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

Example 4(a) – Solution:

Example 4(a) – Solution Because 300  lies in Quadrant IV, the angle it makes with the x -axis is  ′ = 360  – 300  = 60  . Figure 4.40 shows the angle  = 300  and its reference angle  ′ = 60  . Degrees Figure 4.40

Example 4(b) – Solution:

Example 4(b) – Solution Because 2.3 lies between  /2  1.5708 and   3.1416, it follows that it is in Quadrant II and its reference angle is  ′ =  – 2.3  0.8416. Figure 4.41 shows the angle  = 2.3 and its reference angle  ′ =  – 2.3. Radians Figure 4.41 cont’d

Example 4(c) – Solution:

Example 4(c) – Solution First, determine that –135  is coterminal with 225  , which lies in Quadrant III. So, the reference angle is  ′ = 225  – 180  = 45  . Figure 4.42 shows the angle  = –135  and its reference angle  ′ = 45  . Degrees Figure 4.42 cont’d

Slide 18:

Trigonometric Functions of Real Numbers

Trigonometric Functions of Real Numbers:

Trigonometric Functions of Real Numbers To see how a reference angle is used to evaluate a trigonometric function, consider the point ( x , y ) on the terminal side of  , as shown in Figure 4.43. opp = | y |, adj = | x | Figure 4.43

Trigonometric Functions of Real Numbers:

Trigonometric Functions of Real Numbers By definition, you know that sin  = and tan  = . For the right triangle with acute angle  ′ and sides of lengths | x | and | y |, you have sin  ′ = = and tan  ′ = = .

Trigonometric Functions of Real Numbers:

Trigonometric Functions of Real Numbers So, it follows that sin  and sin  ′ are equal, except possibly in sign . The same is true for tan  and tan  ′ and for the other four trigonometric functions. In all cases, the sign of the function value can be determined by the quadrant in which  lies.

Trigonometric Functions of Real Numbers:

Trigonometric Functions of Real Numbers We can greatly extend the scope of exact trigonometric values. For instance, knowing the function values of 30  means that you know the function values of all angles for which 30  is a reference angle.

Trigonometric Functions of Real Numbers:

Trigonometric Functions of Real Numbers For convenience, the table below shows the exact values of the trigonometric functions of special angles and quadrant angles. Trigonometric Values of Common Angles

Example 5 – Using Reference Angles:

Example 5 – Using Reference Angles Evaluate each trigonometric function. a. cos b. tan(–210  ) c. csc

Example 5(a) – Solution:

Example 5(a) – Solution Because  = 4  / 3 lies in Quadrant III, the reference angle is as shown in Figure 4.44. Moreover, the cosine is negative in Quadrant III, so cont’d Figure 4.44

Example 5(b) – Solution:

Example 5(b) – Solution Because –210  + 360  = 150  , it follows that –210  is coterminal with the second-quadrant angle 150  . So, the reference angle is  ′ = 180  – 150  = 30  , as shown in Figure 4.45. cont’d Figure 4.45

Example 5(b) – Solution:

Example 5(b) – Solution Finally, because the tangent is negative in Quadrant II, you have tan( – 210  ) = ( – ) tan 30  = . cont’d

Example 5(c) – Solution:

Example 5(c) – Solution Because (11  / 4) – 2  = 3  / 4, it follows that 11  / 4 is coterminal with the second-quadrant angle 3  / 4. So, the reference angle is  ′ =  – (3  / 4) =  / 4, as shown in Figure 4.46. cont’d Figure 4.46

Example 5(c) – Solution:

Example 5(c) – Solution Because the cosecant is positive in Quadrant II, you have cont’d

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