An angle is formed by two rays that have a common endpoint. You can generate any angle by fixing one ray, called the initial side, and rotating the other ray, called the terminal side, about the vertex.

In a coordinate plane, an angle whose vertex is at the origin and whose initial side is the positive x-axis is in standard position.

The angle measure is positive if the rotation is counterclockwise and negative is clockwise.

More than one rotation is possible.

Draw an angle with the given measure in standard position. Then tell in which quadrant the terminal side lies.

-120o

135o

Find one positive and one negative angle that are coterminal with

-100o

30o

Draw an angle with the given measure in standard position. Then tell in which quadrant the terminal side lies.

-120o Quadrant 3

135o Quadrant 2

Find one positive and one negative angle that are coterminal with

-100o - 100 + 360 = 260 -100 – 360 = - 460

30o 30 + 360 = 390 30 – 360 = -330

To convert degrees to radians,

divide by 180 and multiply by π

To convert radians to degrees,

multiply by 180 and drop the π

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

The values of trigonometric functions of angles greater than90° (or less than 0°) can be found using corresponding acuteangles called reference angles.

Let be an angle in standard position. Its reference angle is the acute angle (read theta prime) formed by the terminal side of and the x-axis.

00'

0

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

00'

0

0'

0

0'

0 = 320°

Because 270° < < 360°, the reference angle is = 360° – 320° = 40°.

00'

Finding Reference Angles

0 = – 56

SOLUTION

Find the reference angle for each angle .00'

Because is coterminal with and < < , the

reference angle is = – = .

076

76

6

76

32

0'

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS

(x, y)

Let be an angle in standard position and (x, y) be any point (except the origin) on the terminal side of . You canFind the reference angle and construct a right triangle.

00

0

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS

(x, y)

Let be the reference angle. Dropping a line from thepoint to the x-axis gives us a right triangle. Now you can find the 6 trigonometric functions.

y

x

ref

ref(x, y)

GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

Pythagorean theorem gives

r = x 2 +y

2.

r

refx

y

In a unit circle the radius is always 1.

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS

sin =0yr

r

y

y

r

csc = , y 00 ry

ref

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS

cos =0xr

r

x

x

r

sec = , x 00 rx

ref

GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS

x

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

y

x

tan = , x 00yx

y

cot = , y 00xy

ref

Evaluating Trigonometric Functions Given a Point

EVALUATING TRIGONOMETRIC FUNCTIONSCONCEPT

SUMMARY

Use these steps to evaluate a trigonometric function ofany angle .0

2

Use the quadrant in which the reference angle liesto determine the sign of the trigonometric functions.

3

1 Find the reference angle

Evaluate the trigonometric function for angle

ref

ref

EVALUATING TRIGONOMETRIC FUNCTIONSCONCEPT

SUMMARY

Evaluating Trigonometric Functions Given a Point

Signs of Function Values

Quadrant IQuadrant II

Quadrant III Quadrant IV

sin , csc : +0 0

tan , cot : +0 0 cos , sec : +0 0

All +Students

Take Calculus

0

Let (3, – 4) be a point on the terminal side of an angle instandard position. Evaluate the six trigonometric functions of .

0

0

SOLUTION

Use the Pythagorean theorem to find the value of r.

r = x 2 + y

2

= 3 2 + (– 4 )

2

= 25

= 5

Evaluating Trigonometric Functions Given a Point

r

(3, – 4)

Using x = 3, y = – 4, and r = 5,you can write the following:

45

sin = = – yr0

cos = =xr

350

tan = = – 0yx

43

csc = = – ry

54

0

sec = =rx

53

0

cot = = – 0xy

34

Evaluating Trigonometric Functions Given a Point

0r

(3, – 4)

Using Reference Angles to Evaluate Trigonometric Functions

Evaluate tan (– 210).

SOLUTION

The angle – 210 is coterminal with 150°.

The tangent function is negative in Quadrant II,so you can write:

tan (– 210) = – tan 30 = – 33

0' = 30

0 = – 210

The reference angle is = 180 – 150 = 30.0'

Using Reference Angles to Evaluate Trigonometric Functions

The cosecant function is positive in Quadrant II,so you can write:

The angle is coterminal with .114

34

2csc = csc = 114

4

0' = 4

0 =11

4

SOLUTION

Evaluate csc .11

4

The reference angle is = – = .

34

4

0'