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

You do not have the permission to view this presentation. In order to view it, please
contact the author of the presentation.

Send to Blogs and Networks

Processing ....

Premium member

Use HTTPs

HTTPS (Hypertext Transfer Protocol Secure) is a protocol used by Web servers to transfer and display Web content securely. Most web browsers block content or generate a “mixed content” warning when users access web pages via HTTPS that contain embedded content loaded via HTTP. To prevent users from facing this, Use HTTPS option.