Trigonometric Ratios in the Unit Circle

14 April 2011

Trigonometric Ratios in the Unit Circle

The unit circle has a radius of 1

tcosr

xtcos

tsinr

ytsin

Trigonometric Ratios in the Unit Circle, cont.

tsecx

rtsec

tcscy

rtcsc

The tangent and cotangent formulas stay the same

“All Students Take Calculus”AS

CT

all ratios are positive

sine is positive

tangent is positive

cosine is positive

cosecant is positive

cotangent is positive

secant is positive

Example:

Trigonometric Ratio

Sine

Cosine

Tangent

Cosecant

Secant

Cotangent

5

Example:

Trigonometric Ratio

Sine

Cosine

Tangent

Cosecant

Secant

Cotangent

18

31

Your Turn:

On the Signs of Trigonometric Ratios handout, complete the feature map and problems 1 – 3

Graphing Negative Radians Find the positive

coterminal angle 1st! Sketch the positive

coterminal angle

6

11

6

112

6

1

6

Graphing Negative Radians, cont.

3

3

12

3

53

5

Graphing Radians with Multiple Revolutions If the angle measure is

larger than 2 pi, keep subtracting 2 from the fraction until the fraction is between 0 and 2 pi. (Find a coterminal angle between 0 and 2 pi.)

3

2

3

22

3

8

3

10

Graphing Multiple Rev. Radians, cont.

5

35

32

5

135

132

5

235

23

Your Turn:

On the Signs of Trigonometric Ratios handout, complete the feature map and problems 4 – 9

The Cardinal Points of the Unit Circle Review

Reminder: Special Right Triangles

23

21 2

2

30°

60°

45°

45°

11

22

30-60-90 45-45-90

Investigation!

Fit the paper triangles onto the picture below. The side with the * must be on the x-axis. Use the paper triangles to determine the coordinates of the three points.

Special Right Triangles & the Unit Circle

Special Right Triangles & the Unit Circle

Bottom half of circle

Special Right Triangles & the Unit Circle

Bottom Half of Circle

Evaluating Trigonometric Expressions

Step 1: Substitute the correct exact value for the trigonometric function. (Use the unit circle!)

Step 2: Evaluate using the order of operations

Examples2

16

sin

Examples 0cos0sin2

sin