r x y z 22 - math.utep.edu · Determine the exact values of the six trigonometric functions of the...

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4.4 Trigonometric Functions of Any Angle Definition of Trig Functions of Any Angle – Let θ be an angle in standard position with (x, y) a point on the terminal side of θ and 2 2 0 r x y . sin cos tan , 0 csc , 0 sec , 0 cot , 0 y x y x r r x r r x y x y y x 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 θ lies. 1. sin 0 cos 0 and 2. sin 0 cos 0 and

Transcript of r x y z 22 - math.utep.edu · Determine the exact values of the six trigonometric functions of the...

Page 1: r x y z 22 - math.utep.edu · Determine the exact values of the six trigonometric functions of the angle. 1. (8, 15) 2. (-4, 10) Examples: State the quadrant in which θ lies. 1.

4.4 Trigonometric Functions of Any Angle

Definition of Trig Functions of Any Angle – Let θ be an angle in standard position with (x, y) a point on

the terminal side of θ and 2 2 0r x y .

sin cos tan , 0

csc , 0 sec , 0 cot , 0

y x yx

r r x

r r xy x y

y x 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 θ lies.

1. sin 0 cos 0and 2. sin 0 cos 0and

Page 2: r x y z 22 - math.utep.edu · Determine the exact values of the six trigonometric functions of the angle. 1. (8, 15) 2. (-4, 10) Examples: State the quadrant in which θ lies. 1.

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

1. 8

cos tan 017with

2. sec 2 sin 0with

Definition of Reference Angle – Let θ be an angle in standard position. Its reference angle is the acute

angle θ’ formed by the terminal side of θ and the horizontal axis.

Page 3: r x y z 22 - math.utep.edu · Determine the exact values of the six trigonometric functions of the angle. 1. (8, 15) 2. (-4, 10) Examples: State the quadrant in which θ lies. 1.

Examples: Find the reference angle and sketch both angles.

1. 309

2. 215

3. 7

6

4. 11.6

Page 4: r x y z 22 - math.utep.edu · Determine the exact values of the six trigonometric functions of the angle. 1. (8, 15) 2. (-4, 10) Examples: State the quadrant in which θ lies. 1.

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

angle θ:

1. Determine the function value for the associated reference angle θ’.

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

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

1. 1

sin2

2. 2 3

csc3

3. tan 1