Trigonometric Ratios in the Unit Circle

Warm-up (2 m)

1. Sketch the following radian measures:

6π17

65

Trigonometric Ratios in the Unit Circle

The unit circle has a radius of 1

θtanxy

θtan

θcosrx

θcos

θsinry

θsin

x is

y is

x is

y is

x is

y is

x is

y is

Quadrant IQuadrant II

Quadrant III Quadrant IV

“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

5π

Example: 18π31

Trigonometric Ratio

Sine

Cosine

Tangent

Your Turn:

Complete problems 1 - 3

Sketching Negative Radians and/or Multiple Revolutions

1. Whenever the angle is less than 0 or more than 2 pi, solve for the coterminal angle between 0 and 2 pi

2. Sketch the coterminal angle

Example #3:3π5

Trigonometric Ratio

Sine

Cosine

Tangent

Example #4: 5π23

Trigonometric Ratio

Sine

Cosine

Tangent

Your Turn:

Complete practice problems 4 – 7

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: 30°- 60°

30°- 60°

45° or 4π

45° or 4π

Summarizing Questions1. In which quadrants is tangent positive?

Why?

2. In which quadrants is cosecant negative? Why?

3. How do I sketch negative angles?

4. How can I sketch angles with multiple revolutions?

5. What are some ways of remembering the radian measures of the Unit Circle?

6. How do we get the coordinates for π/6, π/4, and π/3?

Example #5

43

Example #6

65

Your Turn:

Use your unit circle to solve for the exact values of sine, cosine, and tangent of problems 8 – 11. Rationalize the denominator if necessary.

8.

Sine

Cosine

Tangent

9.

Sine

Cosine

Tangent

3π2

6π

10.

Sine

Cosine

Tangent

11.

Sine

Cosine

Tangent

4π7

2π

Reference Angles

Reference angles make it easier to find exact values of trig functions in the unit circle

Measure an angle’s distance from the x-axis

Reference Angles, cont. Always

Coterminal Acute (less than ) Have one side on the x-axis

2

Solving for Reference Angles Step 1: Calculate the coterminal angle if

necessary (Remember, coterminal angles are positive and less than 2π.)

Step 2: Sketch either the given angle (if less than 2π) or the coterminal angle (if greater than 2π)

Step 3: Determine the angle’s distance from the x-axis (It is almost always pi/denominator!!!)

This is the reference angle!!!!

Example #7:5π6

Example #8:3π2

Example #9:3π7

Your Turn:

4π3

3π4

Your Turn:

6π11

3π4

Your Turn:

3π7

6π17

Your Turn:

5π6

4π7

Your Turn:

4π3

7π

Solving for Exact Trig Values Step 1: Solve for the coterminal angle between

0 and 2π if necessary Step 2: Solve for the reference angle (Note the

quadrant) Step 3: Identify the correct coordinates of the

angle (Make sure the signs of the coordinates match the quadrant!)

Step 4: Solve for the correct trig ratio (Rationalize the denominator if necessary)

Example #10:6π7

Reference Angle:

Coterminal Angle:

Example #10:Coordinates:

Sine:

Tangent:

Cosine:

6π7

Example #11:

Reference Angle:

Coterminal Angle:3π7

Example #11:Coordinates:

Sine:

Tangent:

Cosine:

3π7

Example #12:

Reference Angle:

Coterminal Angle:3π17

Example #12: 3π17

Coordinates:

Sine:

Tangent:

Cosine:

Your Turn:

Complete problems 12 – 18.

Exit Ticket

Solve for the exact values of the following:

1. 2. 3.3π7

sin6π7

cos

2π5

tan

Summarizing QuestionsHow do we get the

coordinates for

using the 45° – 45° – 90°triangle?

Why are the coordinates of negative?

What are the sine, cosine, and tangent of ?

What is a reference angle?

65

65

65

Exit Ticket – “The Important Thing”

On a sheet of paper (with your name!) complete the sentence below:

Three important ideas/things from today’s lesson are ________, ________, and

________, but the most important thing I learned today was ________.