7.4 Trigonometric Functions of General Angles

1. 7.4 Trigonometric Functions of General Angles In this section, we will study the following topics: Evaluating trig functions of any angle Using the unit circle to evaluate the trig functions of quadrantal angles Finding coterminal angles Using reference angles to evaluate trig functions.

2. Trig Functions of Any Angle In 7.3, we looked at the definitions of the trig functions of acute angles of a right triangle. In this section, we will expand upon those definitions to include ANY angle. We will be studying angles that are greater than 90° and less than 0°, so we will need to consider the signsof the trig functions in each of the quadrants. We will start by looking at the definitions of the trig functions of any angle.

3. y (x, y)  r x Definitions of Trig Functions of Any Angle Definitions of Trigonometric Functions of Any Angle Let  be an angle in standard position with (x, y) a point on the terminal side of  and

4. y  x -12 r -5 (-12, -5) Example* Let (-12, -5) be a point on the terminal side of . Find the exact values of the six trig functions of . First you must find the value of r:

5. y  x -12 13 -5 (-12, -5) Example (cont)

6. You Try! Let (-3, 7) be a point on the terminal side of . Find the value of the six trig functions of .

7. The Signs of the Trig Functions Since the radius is always positive (r > 0), the signs of the trig functions are dependent upon the signs of x and y. Therefore, we can determine the sign of the functions by knowing the quadrant in which the terminal side of the angle lies.

8. The Signs of the Trig Functions

9. S A T C A trick to remember where each trig function is POSITIVE: All Students Take Calculus Translation: A = All 3 functions are positive in Quad 1 S= Sine function is positive in Quad 2 T= Tangent function is positive in Quad 3 C= Cosine function is positive in Quad 4 *In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tan is positive, but sine and cosine are negative; ... **Reciprocal functions have the same sign. So cosecant is positive wherever sine is positive, secant is positive wherever cosine is positive, …

10. Example Determine if the following functions are positive or negative: sin 210° cos 320° cot (-135°) csc 500° tan 315°

11. Example* Given and , find the values of the five other trig function of . Solution First, determine the quadrant in which  lies. Since the cosine is negative and the cotangent is positive, we know that  lies in Quadrant _____ .

12. Example* (cont) Now we can find the values of the remaining trig functions:

13. Another Example Given and , find the values of the five other trig functions of .

14. (0, 1) (-1, 0) (1, 0) 0 (0, -1) Trig functions of Quadrantal Angles To find the sine, cosine, tangent, etc. of angles whose terminal side falls on one of the axes , we will use the unit circle. • Unit Circle: • Center (0, 0) • radius = 1 • x2 + y2 = 1

15. Now using the definitions of the trig functions with r = 1, we have:

16. (0, 1) (-1, 0) (1, 0) 0  (0, -1) Example* Find the value of the six trig functions for

17. Example Find the value of the six trig functions for

18. Coterminal Angles Two angles in standard position are said coterminal to be if they have the same terminal sides.  is a negative angle coterminal to   is a positive angle (> 360°) coterminal to  In each of these illustrations, angles  and  are coterminal.

19. Example of Finding Coterminal Angles You can find an angle that is coterminal to a given angle  by adding or subtracting multiples of 360º or 2. Example: Find one positive and one negative angle that are coterminal to 112º. For a positive coterminal angle, add 360º : 112º + 360º = 472º For a negative coterminal angle, subtract 360º: 112º - 360º = -248º Note: There are an infinite number of angles that are coterminal to 112 º.

20. Example • Find one positive and one negative coterminal angle of

22. Reference Angles The values of the trig functions for non-acute angles (Quads II, III, IV) can be found using the values of the corresponding reference angles. I will use the notation to represent an angle’s reference angle.

23. Reference Angles

24. y By sketching  in standard position, we see that it is a 3rd quadrant angle. To find , you would subtract 180° from 225 °.  x Example Find the reference angle for Solution

25. More Examples Find the reference angles for the following angles. 1. 2. 3.

26. So what’s so great about reference angles? • Well…to find the value of the trig function of any non-acute angle, we just need to find the trig function of the reference angle and then determine whether it is positive or negative, depending upon the quadrant in which the angle lies. • For example, In Quad 3, sin is negative 45° is the ref angle

27. Trig Functions of Common Angles Using reference angles and the special reference triangles, we can find the exact values of the common angles. To find the value of a trig function for any common angle  Determine the quadrant in which the angle lies. Determine the reference angle. Use one of the special triangles to determine the function value for the reference angle. Depending upon the quadrant in which  lies, use the appropriate sign (+ or –).

28. More Examples • Give the exact value of the trig function (without using a calculator). • 1. 2.

29. More Examples • Give the exact value of the trig function (without using a calculator). • 3. 4.

30. Evaluating trig function of “uncommon” angles • To find the value of the trig functions of angles that do NOT reference 30°, 45°, or 60°, and are not quadrantal, we will use the calculator. Round your answer to 4 decimal places, if necessary. • Make sure the Mode setting is set to the correct form of the angle: Radian or Degree • To find the trig functions of csc, sec, and cot, use the reciprocal identities.

31. Examples Evaluate the trig functions to four decimal places. 2. 3.

32. End of Section 7.4