An angle is formed by two rays that have a common endpoint. You can generate any angle by fixing one...
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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.

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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

0

0

'

0

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

The values of trigonometric functions of angles greater than

90° (or less than 0°) can be found using corresponding acuteangles called reference angles.


TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

0

0

0

'

'

'

0

0

0


Finding Reference Angles

Find the reference angle for each angle .

0

0

0

0

0

5

= 320°

= –

6

'

'

'

0

0

0

Because 270°< < 360°, the reference angle is

= 360° – 320° = 40°.

Because is coterminal with and  < < , the

reference angle is = – = .

7

7

3

6

6

2

7

6

6

SOLUTION


TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS

Let be an angle in standard position and (x, y) be any point (except the origin) on the terminal side of . You can

Find the reference angle and construct a right triangle.

0

0

0

(x, y)


TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS

(x, y)

y

x

Let be the reference angle. Dropping a line from the

point to the x-axis gives us a right triangle. Now you

can find the 6 trigonometric functions.

(x, y)


TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS

r = x2+y2.

r

y

x

Pythagorean theorem gives

In a unit circle the radius is always 1.


TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS

r

csc = , y 0

sin =

0

0

y

y

r

r

y

y

r


TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS

r

sec = , x 0

cos =

0

0

x

x

r

x

r

x

r


TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS

y

x

tan = , x 0

cot = , y 0

0

0

x

y

y

x

x

y


Evaluating Trigonometric Functions Given a Point

CONCEPT

EVALUATING TRIGONOMETRIC FUNCTIONS

SUMMARY

Use these steps to evaluate a trigonometric function of

any angle .

0

2

3

1

Find the reference angle

Evaluate the trigonometric function for angle

Use the quadrant in which the reference angle lies

to determine the sign of the trigonometric functions.


Evaluating Trigonometric Functions Given a Point

CONCEPT

EVALUATING TRIGONOMETRIC FUNCTIONS

SUMMARY

Quadrant II

Quadrant I

sin , csc : +

0

0

0

0

0

0

Quadrant III

Quadrant IV

cos , sec : +

tan , cot :+

Signs of Function Values

Students

All +

Take

Calculus


Evaluating Trigonometric Functions Given a Point

Let (3, – 4) be a point on the

terminal side of an angle in

standard position. Evaluate the

six trigonometric functions

of .

0

0

0

r=x2+y2

=32+(– 4)2

= 25

r

(3, –4)

SOLUTION

Use the Pythagorean theorem to find the value of r.

= 5


Evaluating Trigonometric Functions Given a Point

r

0

y

5

r

4

csc = = –

0

sin = = –

0

y

r

4

5

3

x

r

5

0

cos = =

0

sec = =

r

5

x

3

y

x

3

4

cot = =–

tan = = –

0

0

x

y

4

3

Using x = 3, y = – 4, and r = 5,

you can write the following:

(3, –4)


Using Reference Angles to Evaluate Trigonometric Functions

=30

=–210

0

'

0

'

0

The reference angle is = 180– 150 = 30.

3

tan (– 210) = – tan 30 = –

3

Evaluate tan (– 210).

SOLUTION

The angle –210 is coterminal with 150°.

The tangent function is negative in Quadrant II,

so you can write:


Using Reference Angles to Evaluate Trigonometric Functions

Evaluate csc.

=

4

=

0

11

11

3

The angle is coterminal with .

'

0

4

4

4

'

0

3

The reference angle is =  – = .

4

4

11

11

4

csc = csc =

2

4

4

SOLUTION

The cosecant function is positive in Quadrant II,

so you can write:


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