1. Trigonometric Functions of Any Angle 4.4

2. 2  Evaluate trigonometric functions of any angle.  Find reference angles.  Evaluate trigonometric functions of real numbers. Objectives

3. 3 Introduction

4. 4 Recall that when using the unit circle to evaluate the value of a trig function, cos θ = x and sin θ = y. What we didn’t point out is that since the radius (hypotenuse) is 1, the trig values are really cos θ = x /1 and sin θ = y /1. So what if the radius (hypotenuse) is not 1?

5. 5 Introduction The definitions of trigonometric functions were restricted to acute angles. In this section, the definitions are extended to cover any angle. When  is an acute angle, the definitions here coincide with those given in the preceding section.

6. 6 Introduction Because r = cannot be zero, it follows that the sine and cosine functions are defined for any real value of . However, when x = 0, the tangent and secant of  are undefined. For example, the tangent of 90  is undefined. Similarly, when y = 0, the cotangent and cosecant of  are undefined.

7. 7 Introduction The previous definitions imply that tan θ and sec θ are not defined when x = 0. So what values of θ are we talking about? They also imply that cot θ and csc θ are not defined when y = 0. So what values of θ are we talking about? or 90 ̊, 270 ̊ or 0 ̊, 180 ̊

8. 8 Example – Evaluating Trigonometric Functions Let (–3, 4) be a point on the terminal side of . Find the sine, cosine, and tangent of . Solution: You can see that x = –3, y = 4, and -3 4

9. 9 Example – Solution So, you have the following. cont’d -3 4 5

10. 10 Trig of Any Angle Let ( x, y ) be a point on the terminal side of an angle θ in standard position with y x

11. 11 Your Turn: Let θ be an angle whose terminal side contains the point (−2, 5). Find the six trig functions for θ.

12. 12 Trig of Any Angle 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.)

13. 13 Trig of Any Angle In a similar manner, you can verify the results shown.

14. 14 4 5 -3 Example: Given sin θ = 4/5 and tan θ < 0, find cos θ and csc θ.

15. 15 Your Turn: Given tan θ = -3/2 and sin θ < 0, find cos θ and csc θ. Quadrant IV – cos positive & csc negative 3 2 r Solution:

16. 16 Reference Angles

17. 17 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. ’

18. 18 Reference Angles The reference angles for  in Quadrants II, III, and IV are shown below.  ′ =  –  (radians)  ′ = 180  –  (degrees)  ′ =  –  (radians)  ′ =  – 180  (degrees)  ′ = 2  –  (radians)  ′ = 360  –  (degrees)

19. 19 Example – Finding Reference Angles Find the reference angle  ′. a.  = 300  b.  = 2.3 c.  = –135 

20. 20 Example (a) – Solution Because 300  lies in Quadrant IV, the angle it makes with the x-axis is  ′ = 360  – 300  = 60 . The figure shows the angle  = 300  and its reference angle  ′ = 60 . Degrees

21. 21 Example (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. The figure shows the angle  = 2.3 and its reference angle  ′ =  – 2.3. Radians cont’d

22. 22 Example (c) – Solution First, determine that –135  is coterminal with 225 , which lies in Quadrant III. So, the reference angle is  ′ = 225  – 180  = 45 . The figure shows the angle  = –135  and its reference angle  ′ = 45 . Degrees cont’d

23. 23 Reference Angles When your angle is negative or is greater than one revolution, to find the reference angle, first find the positive coterminal angle between 0° and 360° or 0 and 2.

24. 24 Your Turn: Find the reference angle for each of the following. 1.213° 2.1.7 rad 3.−144° -144 ̊ is coterminal to 216 ̊ 216 ̊ - 180 ̊ = 36 ̊

25. 25 Trigonometric Functions of Real Numbers

26. 26 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 below. opp = | y |, adj = | x |

27. 27 Trigonometric Functions of Real Numbers How Reference Angles Work: Same except maybe a difference of sign, depending on the quadrant the terminal side of is in.

28. 28 Trigonometric Functions of Real Numbers To find the value of a trig function of any angle: 1.Find the trig value for the associated reference angle. 2.Pick the correct sign depending on where the terminal side lies.

29. 29 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 quadrant in which  lies determines the sign of the function value.

30. 30 Trigonometric Functions of Real Numbers You 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.

31. 31 Trigonometric Functions of Real Numbers For convenience, the table below shows the exact values of the sine, cosine, and tangent functions of special angles and quadrant angles. Trigonometric Values of Common Angles

32. 32 Example – Using Reference Angles Evaluate each trigonometric function. a. cos b. tan(–210  ) c. csc

33. 33 Example (a) – Solution Because  = 4  / 3 lies in Quadrant III, the reference angle is as shown in the figure. Moreover, the cosine is negative in Quadrant III, so

34. 34 Example (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 the figure. cont’d

35. 35 Example (b) – Solution Finally, because the tangent is negative in Quadrant II, you have tan( – 210  ) = ( – ) tan 30  =. cont’d

36. 36 Example (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 the figure. cont’d

37. 37 Example (c) – Solution Because the cosecant is positive in Quadrant II, you have cont’d

38. 38 Your Turn: Evaluate: 1.sin 5/3 2.cos (−60°) 3.tan 11/6

39. 39 Your Turn: Let θ be an angle in Quadrant III such that sin θ = −5/13. Find a) sec θ and b) tan θ using trig identities.

40. 40 Assignment: Pg. 294-296: #1 – 107 odd, 111.