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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.
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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.
–
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Checkpoint Draw Angles in Standard Position
Draw an angle with the given measure in standard position.
ANSWER
30°1. –
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Checkpoint Draw Angles in Standard Position
Draw an angle with the given measure in standard position.
ANSWER
460°2.
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Checkpoint Draw Angles in Standard Position
Draw an angle with the given measure in standard position.
ANSWER
230°3.
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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.
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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°–
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Example 2 Find Coterminal Angles
b. =395° 360°– 35°
=395° 360° – 325°( (2–
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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° –
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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–
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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–
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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
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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,
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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
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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.
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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.
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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°.
=
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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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VOLLEYBALL players spike the ball at speeds up to 100 miles per hour to prevent the opponent from being able to return the ball.
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