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Lesson 12.2

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Your Notes

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**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

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**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

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**Draw Angles in Standard Position**

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

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**Draw Angles in Standard Position**

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

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**Draw Angles in Standard Position**

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

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**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

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**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

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**Find Coterminal Angles**

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

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**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°, ° – 6. 375° ANSWER 15°, ° – 7. 70° – ANSWER 290°, ° –

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**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 , and r = – = r y sin 5 3 – x cos 4 tan 15

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**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

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**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

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**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

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**q q q q q Trigonometric Functions of a Quadrantal Angle Example 4**

Evaluate the sine, cosine, and tangent functions of °. = q SOLUTION When °, you know that x r and y = – q = r y sin q = r x cos – 1 q = x y tan r – q 19

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**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

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**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

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**q Positive and Negative Trigonometric Functions Checkpoint 11.**

Evaluate the sine, cosine, and tangent functions of °. = q ANSWER sin 90° 1, cos 90° , tan 90° is undefined = Determine whether the sine, cosine, and tangent functions of the angle are positive or negative. 12. 40° ANSWER all positive

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**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.

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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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13.3 Evaluating Trigonometric Functions

13.3 Evaluating Trigonometric Functions

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