# sin is an abbreviation for sine cos is an abbreviation for cosine

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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

Which of the following represents r in the figure below
Which of the following represents r in the figure below? (Click on the blue.) x P(x,y) r y Close. The Pythagorean Theorem would be a good beginning but you will still need to “get r alone.” You’re kidding right? (xy)/2 represents the area of the triangle! CORRECT!

Which of the following represents sin  in the figure below
Which of the following represents sin  in the figure below? (Click on the blue.) x P(x,y) r y Sorry. Does Some Old Hippy Caught Another Hippy Tripping on Acid sound familiar? y/x represents tan. Sorry. Does SohCahToa ring a bell? x/r represents cos. CORRECT! Well done.

Which of the following represents cos  in the figure below
Which of the following represents cos  in the figure below? (Click on the blue.) x P(x,y) r y CORRECT! Yeah! Oops! Try something else. Sorry. Wrong ratio.

Which of the following represents tan  in the figure below
Which of the following represents tan  in the figure below? (Click on the blue.) x P(x,y) r y CORRECT! Yeah! Try again. Try again.

In your notes, please copy this figure and the following three ratios:
x P(x,y) r y

A few key points to write in your notebook:
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 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.

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

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.

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

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

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

Students All x Take Calculus

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)

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

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 Check Answer Check Answer

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

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

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:

You try some: Give the exact value in simplest radical form: