1. 7-4 Evaluating Trigonometric Functions of Any Angle Evaluate trigonometric functions of any angle Use reference angles to evaluate trigonometric functions Evaluate trigonometric functions of real numbers

2. 0 P(x,y) r x y A few key points to write in your notebook: P(x,y) can lie in any quadrant. Since the hypotenuse r, represents distance, the value of r is always positive. The equation x 2 + y 2 = r 2 represents the equation of a circle with its center at the origin and a radius of length r. The trigonometric ratios still apply but you will need to pay attention to the +/– sign of each. Recall In General

3. Example: If the terminal ray of an angle  in standard position passes through (–3, 2), find sin  and cos . You try this one in your notebook: If the terminal ray of an angle  in standard position passes through (–3, –4), find sin  and cos . (–3,2) r –3 2 Check Answer

4. Example: If  is a fourth-quadrant angle and sin  = –5/13, find cos . 13 –5 x Since  is in quadrant IV, the coordinate signs will be (+x, –y), therefore x = +12.

5. Example: If  is a second quadrant angle and cos  = –7/25, find sin . Check Answer

6. x 0 P(–x,y) r y 0 P(–x, –y) r x y P(x,y) 0 r x y 0 P(x, –y) r x y Determine the signs of sin , cos , and tan  according to quadrant. Quadrant II is completed for you. Repeat the process for quadrants I, III, and IV. Hint: r is always positive; look at the red P coordinate to determine the sign of x and y.

7. y x AllSine TangentCosine Check your answers according to the chart below: All are positive in I. Only sine is positive in II. Only tangent is positive in III. Only cosine is positive in IV.

8. y x AllStudents TakeCalculus A handy pneumonic to help you remember! Write it in your notes!

9. x 0 P(–x,y) r y 0 P(–x, –y) r x y P(x,y) 0 r x y 0 P(x, –y) r x y Let  be an angle in standard position. The reference angle  associated with  is the acute angle formed by the terminal side of  and the x-axis. 1.Find the reference angle α. 2.Determine the sign by noting the quadrant. 3.Evaluate and apply the sign.

10. Example: Find the reference angle for  = 135 . You try it: Find the reference angle for  = 5  /3. You try it: Find the reference angle for  = 870 . Check Answer

11. Give each of the following in terms of the cosine of a reference angle: Example: cos 160  The angle  =160  is in Quadrant II; cosine is negative in Quadrant II (refer back to All Students Take Calculus pneumonic). The reference angle in Quadrant II is as follows:  =180 –  or  =180 – 160 = 20. Therefore: cos 160  = –cos 20  You try some: cos 182  cos (–100  ) cos 365  Check Answer

12. Try some sine problems now: Give each of the following in terms of the sine of a reference angle: sin 170  sin 330  sin (–15  ) sin 400  Check Answer

13. Give the exact value in simplest radical form. Example: sin 225  Determine the sign: This angle is in Quadrant III where sine is negative. Find the reference angle for an angle in Quadrant III:  =  – 180  or  = 225  – 180  = 45 . Therefore:

14. You try some: Give the exact value in simplest radical form: sin 45  sin 135  sin 225  cos (–30  ) cos 330  sin 7  /6 cos  /4 Check Answer

15. Not all angles are SPECIAL. Sometimes you need to use your calculator. Be careful. Some problems are in degrees and some problems are in radians. Either switch back and forth between the two modes in your calculator. Or keep it in degree mode and convert quickly from radians to degrees first... sort of anyway. Example: sin 217  Make sure you are in DEGREE mode and just type it in your calculator! The angle  =217  is in Quadrant III; sin is negative in Quadrant III, so the sign of the angle makes sense.

16. You try some: Give the value rounded to 3 places: sin 28  cos 238  tan 302  cos (–15  ) sin  /9 cos (–2  /5) tan 15  /7 Check Answer

17. Homework: Page 294-296, #5-7, 13-16, 17, 19, 21, 38, 40, 42, 44, 81-90