R x y z 22 - UTEP

4.4 Trigonometric Functions of Any Angle

Definition of Trig Functions of Any Angle ¨C Let ¦È be an angle in standard position with (x, y) a point on

the terminal side of ¦È and r ?

x2 ? y 2 ? 0 .

y

x

cos ? ?

tan ? ?

r

r

r

r

csc ? ? , y ? 0 sec ? ? , x ? 0 cot ? ?

y

x

sin ? ?

y

, x?0

x

x

, y?0

y

Examples: The point is on the terminal side of an angle in standard position. Determine the exact values

of the six trigonometric functions of the angle.

1. (8, 15)

2. (-4, 10)

Examples: State the quadrant in which ¦È lies.

1. sin ? ? 0 and cos? ? 0

2. sin ? ? 0 and cos? ? 0

Example: Find the values of the six trig functions of ¦È with the given constraint.

1. cos ? ?

8

with tan ? ? 0

17

2. sec? ? ?2 with sin ? ? 0

Definition of Reference Angle ¨C Let ¦È be an angle in standard position. Its reference angle is the acute

angle ¦È¡¯ formed by the terminal side of ¦È and the horizontal axis.

Examples: Find the reference angle and sketch both angles.

1. ? ? 309

2. ?215

3. ? ?

7?

6

4. ? ? 11.6

Evaluating Trigonometric Functions of Any Angle ¨C To find the value of a trigonometric function of any

angle ¦È:

1. Determine the function value for the associated reference angle ¦È¡¯.

2. Depending on the quadrant in which ¦È lies, affix the appropriate sign to the function value.

Examples: Find two solutions of the equation. Give your answers in degrees and radians.

1. sin ? ?

1

2

2. csc ? ?

2 3

3

3. tan ? ? 1

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