Your Notes Example 1 Draw Angles in Standard Position Draw an angle with the given measure in standard position. 215° a. 410° b. 60° c. – SOLUTION.

Example 1 Draw Angles in Standard Position

Draw an angle with the given measure in standard position.

215°a. 410°b. 60°c. –

SOLUTION

a. Because 215° is 35° more than 180°, the terminal side is 35° counterclockwisepast the negative x-axis.

Example 1 Draw Angles in Standard Position

b. Because 410° is 50° more than 360°, the terminal side makes one whole revolution counterclockwise plus 50° more.

c. Because 60° is negative, the terminal side is 60° clockwise from the positive x-axis.

–

Checkpoint Draw Angles in Standard Position

Draw an angle with the given measure in standard position.

ANSWER

30°1. –

Checkpoint Draw Angles in Standard Position

Draw an angle with the given measure in standard position.

ANSWER

460°2.

Checkpoint Draw Angles in Standard Position

Draw an angle with the given measure in standard position.

ANSWER

230°3.

Checkpoint Draw Angles in Standard Position

Draw an angle with measure 90° in standard position. On a different coordinate grid, draw an angle with measure 90° not in standard position.

ANSWER

4.

Example 2 Find Coterminal Angles

SOLUTION

There are many correct answers. Choose a multiple of 360° to add or subtract.

Find one positive angle and one negative angle that are coterminal with the given angle.

a. b. 395°45°–

a. =45°– 360°+ 315°

=45°– 360°– 405°–

Example 2 Find Coterminal Angles

b. =395° 360°– 35°

=395° 360° – 325°( (2–

Checkpoint

Find one positive angle and one negative angle that are coterminal with the given angle.

5. 50°

Find Coterminal Angles

ANSWER 410°, 310° –

6. 375°

ANSWER 290°, 430° –7. 70°–

ANSWER 15°, 345° –

Example 3 Evaluate Trigonometric Functions Given a Point

SOLUTION

Use the Pythagorean theorem to find the value of r.

Let be a point on the terminal side of an angle in standard position. Evaluate the sine, cosine, and tangent functions of .

( )4, – 3

=r = 42 + ( )23–x 2 y 2+ = 25 = 5

Find the value of each function using x 4, y 3, and r 5.= = – =

=r

ysin =

5

3– =

r

xcos =

5

4=

x

ytan =

4

3–

CheckpointEvaluate Trigonometric Functions Given a Point

ANSWER

=sin5

4, =cos

5

3– , =tan

3

4–

8.

Use the given point on the terminal side of an angle in standard position. Evaluate the sine, cosine, and tangent functions of .

( )3, 4–

CheckpointEvaluate Trigonometric Functions Given a Point

ANSWER

=sin5

4, =cos

5

3, =tan

3

4

9.

Use the given point on the terminal side of an angle in standard position. Evaluate the sine, cosine, and tangent functions of .

( )6, 8

CheckpointEvaluate Trigonometric Functions Given a Point

ANSWER

=sin , =cos ,17

8–

17

15=tan

8

15–

10.

Use the given point on the terminal side of an angle in standard position. Evaluate the sine, cosine, and tangent functions of .

–( )158,

Example 4 Trigonometric Functions of a Quadrantal Angle

Evaluate the sine, cosine, and tangent functions of 180°.=

SOLUTION

When 180°, you know that x r and y 0. = –= =

=r

ysin =

r

0= 0

=r

xcos =

r

r–= 1–

=x

ytan =

r

0

–= 0

Example 5 Positive and Negative Trigonometric Functions

Determine whether the sine, cosine, and tangent functions of the given angle are positive or negative.

a. b.

c. d.

Example 5 Positive and Negative Trigonometric Functions

SOLUTION

Because the terminal side lies in Quadrant II, sin 100° is positive, cos 100° is negative, and tan 100° is negative.

a.

Because the terminal side lies in Quadrant I, sin 75° is positive, cos 75° is positive, and tan 75° is positive.

b.

Because the terminal side lies in Quadrant III, sin 210° Is negative, cos 210° is negative, and tan 210° is positive.

c.

Because the terminal side lies in Quadrant IV, sin 320° is negative, cos 320° is positive, and tan 320° is negative.

d.

CheckpointPositive and Negative Trigonometric Functions

ANSWER sin 90° 1, cos 90° 0, tan 90° is undefined= =

Determine whether the sine, cosine, and tangent functions of the angle are positive or negative.

12. 40°

ANSWER all positive

11. Evaluate the sine, cosine, and tangent functions of 90°.

=

CheckpointPositive and Negative Trigonometric Functions

Determine whether the sine, cosine, and tangent functions of the angle are positive or negative.

13. 150°

ANSWER

The sine is positive, the cosine is negative, and the tangent is negative.

14. 225°

ANSWER

The sine is negative, the cosine is negative, and thetangent is positive.

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