7 4 Evaluating Trigonometric Functions of Any Angle

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

Recall y P(x, y) In General r 0 x 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.

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 (– 3, 2) r 2 – 3

Example: If is a fourth-quadrant angle and sin = – 5/13, find cos . x 13 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

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. y y P(–x, y) P(x, y) r r x 0 y r 0 P(–x, –y) x 0 y x 0 x r P(x, –y)

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 Sine All x Tangent Cosine

A handy pneumonic to help you remember! Write it in your notes! y Students All x Take Calculus

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 x 0 1. Find the reference angle α. 2. Determine the sign by noting the quadrant. y 0 r P(–x, –y) 3. Evaluate and apply the sign. y x 0 x r P(x, –y)

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

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 Check Answer • cos (– 100 ) Check Answer • cos 365 Check Answer

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

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: • sin 45 Check Answer • sin 135 Check Answer • sin 225 Check Answer • cos (– 30 ) Check Answer • cos 330 Check Answer • sin 7 /6 Check Answer • cos /4 Check Answer

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.

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

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