reference angles and trigonometry using trigonometry in a right triangle we were limited to acute...

Reference AnglesAnd Trigonometry

Using Trigonometry in a Right TriangleWe were limited to Acute Angles

We can extend Trigonometry to Angles of Any Measureby placing those angles in the coordinate plane

We do this by using reference angles,Acute Angles measured to the x-axis.

Angles are Placed with one sidecalled the initial side on the positive x-axis.

The terminal side is rotated counter-clockwise.

135°

A Reference Angle is measuredto the x-axis.

The terminal side is rotated counter-clockwise.

135°

45°

A Reference Angle is measuredto the x-axis.

The terminal side is rotated counter-clockwise.

225°

45°

A Reference Angle is measuredto the x-axis.

The terminal side is rotated counter-clockwise.

315°

45°

A Reference Angle is measuredto the x-axis.

If the terminal side is rotated clockwise, the angle measure isNegative.

-45°

45°

Always Positive.

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

Unit Circle has a radius

of 1 unit.

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

Unit Circle has a radius

of 1 unit.

45°

45°

1

2

2

2

2

Cos

+

Sin

+

2

2

2

2

x=

=y

Cosine = xSine = y

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

45°45°

Cos

+

Sin

+

135°

Reference Angle =

Cos

-

Sin

+

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

45°

45°

Cos

+

Sin

+

225°

Reference Angle =

Cos

-

Sin

+

Cos

-

Sin

-45°

135°45°

225°

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

45°

45°

Cos

+

Sin

+

315°

Reference Angle

Cos

-

Sin

+

Cos

-

Sin

-45°

135°45°

225°

45°

Cos

+

Sin

-

315°

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

45°

45°

Cos

+

Sin

+Cos

-

Sin

+

Cos

-

Sin

-45°

135°45°

225°

45°

Cos

+

Sin

-

315°

Quadrant 2Quadrant 1

Quadrant 4Quadrant 3

, 180 0, 02, 360

3

2

2

1,0

0,1

45°

Cos

+

Sin

+

45°

Quadrant 1

Cosine = xSine = y

Tangent = Δy Δx

Tangent = Sine Cosine

1

22

22

tan

tan = 1

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

45°

45°

Cos

+

Sin

+Cos

-

Sin

+

Cos

-

Sin

-45°

135°45°

225°

45°

Cos

+

Sin

-

315°

Quadrant 2Quadrant 1

Quadrant 4Quadrant 3

tan = 1tan = -1

tan = -1tan = 1

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

Cos

+

Sin

+Cos

-

Sin

+

Cos

-

Sin

-

Cos

+

Sin

-

Quadrant 2Quadrant 1

Quadrant 4Quadrant 3

Tan

-

Tan

-T

an +

Tan

+

Tangent = Sine Cosine

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

30°

Cos

+

Sin

+

30°

1

2

1

2

3

2

1

2

3

Cosine = xSine = y

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

30°30°

Cos

+

Sin

+Cos

-

Sin

+

Cos

-

Sin

-30°

150°30°

210°

30°

Cos

+

Sin

-

330°

150°

, 180 0, 02, 360

3

2

2

Cos

+

Sin

+Cos

-

Sin

+

Cos

-

Sin

-

150°30°

210°

Cos

+

Sin

-

330°

3

3

3

1

3

2

2

1

23

21

)30tan(

3

3)30tan( 3

3)150tan(

3

3)270tan(

3

3)330tan(

Tangent = Sine Cosine

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

60°

Cos

+

Sin

+

60°

1

2

1

2

32

1

2

3

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

60°

Cos

+

Sin

+

60°

2

1

2

3

60°

120°

Cos

+

Sin

+

120°

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

60°

Cos

+

Sin

+

60°

2

1

2

3

60°

Cos

-

Sin

+

120°

60°

Cos

-

Sin

-

240°

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

60°

Cos

+

Sin

+

60°

60°

Cos

-

Sin

+

120°

60°

Cos

-

Sin

-

240°

60°

Cos

+

Sin

-300°

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1 C

os +

Sin

+

60°

Cos

-

Sin

+

120°

Cos

-

Sin

-

240°C

os +

Sin

-300°

31

2

2

3

21

23

)60tan(

Tangent = Sine Cosine

3)240tan(

3)120tan( 3)60tan(

3)300tan(

, 180 0, 02, 360

3

2

2

Cosine = xSine = y

Cos

+

Sin

(1 , 0)(-1 , 0)

(0 , 1)

(0 , -1)

Cos(0) =1Sin(0) = 0

Cos

Sin

+ Cos(90) =0Sin(90) = 1

90°180°

Cos

-

Sin

Cos(180) = -1Sin(180) = 0

270°

Cos

Sin

- Cos(270) = 0

Sin(270) = -1

, 180 0, 02, 360

3

2

2

Cosine = xSine = y

Tangent = Sine Cosine

Cos

+

Sin

(1 , 0)(-1 , 0)

(0 , 1)

(0 , -1)

Cos(0) =1Sin(0) = 0

Cos

Sin

+ Cos(90) =0Sin(90) = 1

90°180°

Cos

-

Sin

Cos(180) = -1Sin(180) = 0

270°

Cos

Sin

- Cos(270) = 0

Sin(270) = -1

Tan(0) =0

Tan(90) undefined

Tan(180) =0

Tan(270) undefined

Evaluate the trigonometric functions at each real number.

2

3,

2

1

3

2Sin

3

2Cos

3

2Tan

= y

= x

x

y

2

3

2

1

2

1

2

3

1

2

2

3

3

1203

2

Evaluate the six trigonometric functions at each real number.

(0, -1)2

2Sin

2Cos

2Tan

= y

= x

= -1

= 0

x

y

0

1DNE

Does Not Exist

2Sec

0

1 DNE

2Cot

1

0

2Csc

= -1

= 0

Evaluate the six trigonometric functions at each real number.

4

7

2

2,

2

2

Sin

4

7

4

7Cos

4

7Tan

4

7Csc

4

7Sec

4

7Cot

2

2

2

2

-1-1

2

2

So, you think you got it now?