y x radian: the length of the arc above the angle divided by the radius of the circle. definition,...

-1 1

-1

1

y

x

Radian: The length of the arc above the angle divided by the radius of the circle.

Definition

sr

s

r , in radians

-1 1

-1

1

y

x

Definition

s

1

s , in radians

Unit Circle: the circle with radius of 1 unitIf r=1, =s

1

DefinitionThe radian measure of an angle is the distance traveled around the unit circle. Since circumference of a circle is 2 r and r=1, the distance around the unit circle is 2

Important IdeaIf a circle contains 360° or 2 radians, how many radians are in 180°

• Use to change rads to degrees

180° rad

s• Use to change

degrees to rads rad

s180°

Try This

Change 240° to radian measure in terms of .

4 rads

3

Try This

Change radians to

degree measure.

157.5°

7

8

Try This

Change radians to

degree measure.

171.89°

3

Definition

Initial Side

Terminal Side

Vertex Angle A

is in standard position

A x

y

Definition

A

If the terminal side moves counter-clockwise, angle A is positive

x

y

Definition

A

If the terminal side moves counter-clockwise, angle A is positive

x

y

Definition

A

If the terminal side moves counter-clockwise, angle A is positive

x

y

Definition

A

If the terminal side moves clockwise, angle A is negative

x

y

Definition

A

If the terminal side moves clockwise, angle A is negative

x

y

Definition

A

If the terminal side moves clockwise, angle A is negative

x

y

Definition

A

If the terminal side moves clockwise, angle A is negative

x

y

Definition

A

If the terminal side is on an axis, angle A is a quadrantel angle

x

y

Definition

A

If the terminal side is on an axis, angle A is a quadrantel angle

x

y

Definition

A

If the terminal side is on an axis, angle A is a quadrantel angle

x

y

Definition

A

If the terminal side is on an axis, angle A is a quadrantel angle

x

y

Definition

0

2

32

The quadrantal angles in radians 2

Definition

0

2

32

The quadrantal angles in radians 2

Definition

0

2

32

The quadrantal angles in radians 2

Definition

0

2

The quadrantal angles in radians 2

The terminal side is on an axis.

DefinitionCoterminal Angles: Angles that have the same terminal side.

Important IdeaIn precal, angles can be larger than 360° or 2 radians.

Important IdeaTo find coterminal angles, simply add or subtract either 360° or 2 radians to the given angle or any angle that is already coterminal to the given angle.

Analysis30° and 390° have the same terminal side, therefore, the angles are coterminal

30°x

y

x

y390°

Analysis30° and 750° have the same terminal side, therefore, the angles are coterminal

30°x

y

x

y750°

Analysis30° and 1110° have the same terminal side, therefore, the angles are coterminal

30°x

y

x

y 1110°

Analysis30° and -330° have the same terminal side, therefore, the angles are coterminal

30°x

y

x

y -330°

Try ThisFind 3 angles coterminal with 60°

420°,780° and -300°

Try ThisFind two positive angle and one negative angle coterminal with radians.

56

and 76

196

176

,

Important Idea

( , )x y

r

x

y

opp

cos x

r

hyp

sin y

rhyp

adj

tan oppadj

y

x

r > 0

Find sin, cos & tan of the angle whose terminal side passes through the point (5,-12)

Try This

(5,-12)

Solution

5

-121

3

12sin

13

5cos

13

12tan

5

(5,-12)

Important IdeaTrig ratios may be positive or negative

Find the exact value of the sin, cos and tan of the given angle in standard position. Do not use a calculator.

11

6

Solution 11

6

-13

2

11 1sin

6 2

11 3

cos6 2

11 1 3tan

6 33

DefinitionReference Angle: the acute angle between the terminal side of an angle and the x axis. (Note: x axis; not y axis). Reference angles are always positive.

Important IdeaHow you find the reference angle depends on which quadrant contains the given angle.

The ability to quickly and accurately find a reference angle is going to be important in future lessons.

ExampleFind the reference angle if the given angle is 20°.

In quad. 1, the given angle & the ref. angle are the same.

x

y

20°

ExampleFind the reference angle if the given angle is 120°.For given

angles in quad. 2, the ref. angle is 180° less the given angle.

?120°x

y

ExampleFind the reference angle if the given angle is .

x

y

7

6

7

6

For given angles in quad. 3, the ref. angle is the given angle less

Try ThisFind the reference angle if the given angle is

7

4

For given angles in quad. 4, the ref. angle is less the given angle.

2

7

4

4

Important IdeaThe trig ratio of a given angle is the same as the trig ratio of its reference angle except, possibly, for the sign.

Example: sin 130 sin 50

sin 230 sin 50

The unit circle is a circle with radius of 1. We use the unit circle to find trig functions of quadrantal angles.

-1 1

-1

1

1

Definition

The unit circle

-1 1

-1

1

1

Definition

(1,0)

(0,1)

(-1,0)

(0,-1)

x y

Definition

-1 1

-1

1

(1,0)

(0,1)

(-1,0)

(0,-1)

For the quadrantal angles:

The x values are the terminal sides for the cos function.

Definition

-1 1

-1

1

(1,0)

(0,1)

(-1,0)

(0,-1)

For the quadrantal angles:

The y values are the terminal sides for the sin function.

Definition

-1 1

-1

1

(1,0)

(0,1)

(-1,0)

(0,-1)

For the quadrantal angles :

The tan function is the y divided by the x

-1 1

-1

1

Find the values of the 6 trig functions of the quadrantal angle in standard position:

Example

sincostan

cscseccot0°

(1,0)

(0,1)

(-1,0)

(0,-1)

-1 1

-1

1Find the values of the 6 trig functions of the quadrantal angle in standard position:

Example

sincostan

cscseccot90

°

(1,0)

(0,1)

(-1,0)

(0,-1)

-1 1

-1

1Find the values of the 6 trig functions of the quadrantal angle in standard position:

Example

sincostan

cscseccot180°

(1,0)

(0,1)

(-1,0)

(0,-1)

-1 1

-1

1Find the values of the 6 trig functions of the quadrantal angle in standard position:

Example

sincostan

cscseccot270°

(1,0)

(0,1)

(-1,0)

(0,-1)

-1 1

-1

1Find the values of the 6 trig functions of the quadrantal angle in standard position:

Try This

sincostan

cscseccot360°

(1,0)

(0,1)

(-1,0)

(0,-1)

A trigonometric identity is a statement of equality between two expressions. It means one expression can be used in place of the other.A list of the basic identities can be found on p.317 of your text.

1sin

csc

1csc

sin

1cos

sec

1sec

cos

1cot

tan

1tan

cot

Reciprocal Identities:

coscot

sin

AA

A

sintan

AA

cosA

Quotient Identities:

-1 1

-1

1

x

y

r

siny

r

cosx

r

but…2 2 2x y r

therefore

2 2sin cos 1

2 2sin cos 1

Divide by to get:

2cos 2 2tan 1 sec

Pythagorean Identities:

2 2sin cos 1

Pythagorean Identities:

Divide by to get:

2sin 2 21 cot csc

Try ThisUse the Identities to simplify the given expression:

2 2 2cot sin sint t t1

Try ThisUse the Identities to simplify the given expression:

2 2

2

sec tan

cos

t t

t

2sec t

Prove that this is an identity2sin

1 cos1 cos

sin 1 cos2cot sec

1 cos sin

q qq q

q q+

+ =+

Now prove that this is an identity

One More21 1

2secsin 1 sin 1

xx x

- =-- +