1. 7-3 Sine and Cosine (and Tangent) Functions 7-4 Evaluating Sine and Cosine
sin is an abbreviation for sine
cos is an abbreviation for cosine
tan is an abbreviation for tangent

2. Which of the following represents r in the figure below
Which of the following represents r in the figure below? (Circle the correct answer.) x
P(x,y) r y

3. Which of the following represents sin  in the figure below
Which of the following represents sin  in the figure below? (Circle the correct answer.) x
P(x,y) r y

4. Which of the following represents cos  in the figure below
Which of the following represents cos  in the figure below? (Circle the correct answer.) x
P(x,y) r y

5. Which of the following represents tan  in the figure below
Which of the following represents tan  in the figure below? (Circle the correct answer.) x
P(x,y) r y

6. You will need to memorize the following three ratios:x
P(x,y) r y

7. P(x,y) can lie in any quadrant.
P(x,y) r x y
A few key points:
P(x,y) can lie in any quadrant.
Since the hypotenuse r, represents distance, the value of r is always positive.
The equation x2 + y2 = r2 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.

8. (–3,2) r –3 2
Example: If the terminal ray of an angle  in standard position passes through (–3, 2), find sin  and cos .
You try this one: If the terminal ray of an angle  in standard position passes through (–3, –4), find sin  and cos .

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

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

11. Determine the signs of sin  , cos  , and tan  according to quadrant
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.
P(–x,y) r y r x y
P(x,y) x
P(–x, –y) r x y
P(x, –y) r x y

12. 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. y x
All
Sine
Tangent
Cosine

13. A handy pneumonic to help you remember! Memorize it!
Students
All x
Take
Calculus

14. Find the reference angle. Determine the sign by noting the quadrant.
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. y y
P(–x,y)
P(x,y) r r
Find the reference angle.
Determine the sign by noting the quadrant.
Evaluate and apply the sign. x x y
P(x, –y) r x y x r
P(–x, –y)

15. 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.

16. 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

17. Try some sine problems now: Give each of the following in terms of the sine of a reference angle:

18. Can you complete this chart?
60
30
45
60
30
45

19. 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:

20. 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