Holt McDougal Algebra 2

The Unit CircleThe Unit Circle

Holt Algebra 2Holt McDougal Algebra 2

• How do we convert angle measures between degrees and radians?

• How do we find the values of trigonometric functions on the unit circle?

Holt McDougal Algebra 2

The Unit Circle

You can use reference angles and Quadrant I of the unit circle to determine the values of trigonometric functions.

Trigonometric Functions and Reference Angles

Holt McDougal Algebra 2

The Unit Circle

The diagram shows how the signs of the trigonometric functions depend on the quadrant containing the terminal side of θ in standard position.

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Holt McDougal Algebra 2

The Unit Circle

sin

cos

tan

0 30 54 60 90

20

1

0

21

23

33

22

22

1

23

21

3

24

0

und.

0 6 4 3 2radian

0 21 1

Holt McDougal Algebra 2

The Unit CircleUsing Reference Angles to Evaluate Trigonometric Functions

Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle.

Step 1 Find the reference angle.

o330 1.330360 o30Step 2 Find the sin, cos, and tan of the

reference angle. Step 3 Adjust the signs, if needed.

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21330sin o

23330cos o

33330tan o

Holt McDougal Algebra 2

The Unit CircleUsing Reference Angles to Evaluate Trigonometric Functions

Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle.

Step 1 Find the reference angle.

o270 2.

270360 o90Step 2 Find the sin, cos, and tan of the

reference angle. Step 3 Adjust the signs, if needed.

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Take Calculus

270°

tan 90° = und.

sin 90° = 1cos 90° = 0

1270sin o

0270cos o

und.270tan o

Holt McDougal Algebra 2

The Unit CircleUsing Reference Angles to Evaluate Trigonometric Functions

Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle.

Step 1 Find the reference angle.6

11 3.

6112

6

Step 2 Find the sin, cos, and tan of the reference angle. Step 3 Adjust the signs, if needed.

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Take Calculus

21

6sin

23

6cos

33

6tan

21

611sin

23

611cos

33

611tan

Holt McDougal Algebra 2

The Unit CircleUsing Reference Angles to Evaluate Trigonometric Functions

Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle.

Step 1 Find the reference angle.3

4 4.

34

3

Step 2 Find the sin, cos, and tan of the reference angle. Step 3 Adjust the signs, if needed.

AllStudents

Take Calculus

23

3sin

21

3cos

33

tan

34

23

34sin

21

34cos

33

4tan

Holt McDougal Algebra 2

The Unit CircleIf you know the measure of a central angle of a circle, you can determine the length s of the arc intercepted by the angle.

rs

Holt McDougal Algebra 2

The Unit CircleAutomobile Application

5. A tire of a car makes 653 complete rotations in 1 min. The diameter of the tire is 0.65 m. To the nearest meter, how far does the car travel in 1 s?Step 1 Find the radius of the tire.

Step 2 Find the angle θ in radian through which the tire rotates in 1 second.

The radius is of the diameter.

1 rotation = 2.

min 1653

12

Change to seconds sec 06

min 1

sec 30653

30

sec 30653m 325. 2.22rs m/sec

Holt McDougal Algebra 2

The Unit CircleAutomobile Application

6. An minute hand on Big Ben’s Clock Tower in London is 14 ft long. To the nearest tenth of a foot, how far does the tip of the minute hand travel in 1 minute?Step 1 Find the radius of the clock.Step 2 Find the angle θ in radian through which the hour hand rotates in 1 minute.

1 hour = 2.

hour 12

Change to minutesmin 06hour 1

min 30

30

min 30ft 41 5.1rs ft/min

r =14

Holt McDougal Algebra 2

The Unit Circle

Lesson 10.3 Practice B