Lesson 12.2

Your Notes

Draw Angles in Standard Position 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° counterclockwise past the negative x-axis. 6

Draw Angles in Standard Position 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. – 7

Draw Angles in Standard Position Checkpoint Draw Angles in Standard Position Draw an angle with the given measure in standard position. 30° 1. – ANSWER

Draw Angles in Standard Position Checkpoint Draw Angles in Standard Position Draw an angle with the given measure in standard position. 460° 2. ANSWER

Draw Angles in Standard Position Checkpoint Draw Angles in Standard Position Draw an angle with the given measure in standard position. 230° 3. ANSWER

Draw Angles in Standard Position Checkpoint Draw Angles in Standard Position 4. 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

Find Coterminal Angles Example 2 Find Coterminal Angles Find one positive angle and one negative angle that are coterminal with the given angle. a. 45° – b. 395° SOLUTION There are many correct answers. Choose a multiple of 360° to add or subtract. a. = 45° – 360° + 315° 405° 12

Find Coterminal Angles Example 2 Find Coterminal Angles b. 395° – 360° 35° = 395° – 2 ( 360° ( 325° = – 13

Find Coterminal Angles Checkpoint Find Coterminal Angles Find one positive angle and one negative angle that are coterminal with the given angle. 5. 50° ANSWER 410°, 310° – 6. 375° ANSWER 15°, 345° – 7. 70° – ANSWER 290°, 430° –

Evaluate Trigonometric Functions Given a Point Example 3 Evaluate Trigonometric Functions Given a Point Let be a point on the terminal side of an angle in standard position. Evaluate the sine, cosine, and tangent functions of . ( ) 4, – 3 SOLUTION Use the Pythagorean theorem to find the value of r. = r 42 + ( )2 3 – x 2 y 2 25 5 Find the value of each function using x 4, y 3, and r 5. = – = r y sin 5 3 – x cos 4 tan 15

Evaluate Trigonometric Functions Given a Point Checkpoint Evaluate Trigonometric Functions Given a Point 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 – ANSWER = sin 5 4 , cos 3 – tan

Evaluate Trigonometric Functions Given a Point Checkpoint Evaluate Trigonometric Functions Given a Point 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 ANSWER = sin 5 4 , cos 3 tan

Evaluate Trigonometric Functions Given a Point Checkpoint Evaluate Trigonometric Functions Given a Point 10. Use the given point on the terminal side of an angle in standard position. Evaluate the sine, cosine, and tangent functions of . – ( ) 15 8, ANSWER = sin , cos 17 8 – 15 tan

q q q q q Trigonometric Functions of a Quadrantal Angle Example 4 Evaluate the sine, cosine, and tangent functions of 180°. = q SOLUTION When 180°, you know that x r and y 0. = – q = r y sin q = r x cos – 1 q = x y tan r – q 19

Positive and Negative Trigonometric Functions 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. 20

Positive and Negative Trigonometric Functions 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. 21

q Positive and Negative Trigonometric Functions Checkpoint 11. Evaluate the sine, cosine, and tangent functions of 90°. = q 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

Positive and Negative Trigonometric Functions Checkpoint Positive 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 the tangent is positive.

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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