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13.3 Evaluating Trigonometric Functions

13.3 Evaluating Trigonometric Functions. Evaluating Trigonometric Functions Given a Point. Let ( 3 , – 4 ) be a point on the terminal side of an angle in standard position. Evaluate the six trigonometric functions of . 0. 0. 0. r = x 2 + y 2. = 3 2 + ( – 4 ) 2. = 25. r.

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13.3 Evaluating Trigonometric Functions

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  1. 13.3 Evaluating Trigonometric Functions

  2. Evaluating Trigonometric Functions Given a Point Let (3, – 4) be a point on the terminal side of an angle in standard position. Evaluate the six trigonometric functions of . 0 0 0 r=x2+y2 =32+(– 4)2 = 25 r (3, –4) SOLUTION Use the Pythagorean theorem to find the value of r. = 5

  3. Evaluating Trigonometric Functions Given a Point r 0 y r 4 5 csc = = – 0 sin = = – 0 y r 5 4 3 x r 5 0 cos = = sec = = 0 r x 5 3 y x 3 4 cot = =– tan = = – 0 0 x y 4 3 Using x = 3, y = – 4, and r = 5, you can write the following: (3, –4)

  4. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 0 0 ' 0 Let be an angle in standard position. Its reference angleis the acute angle (read thetaprime) formed by the terminal side of and the x-axis. The values of trigonometric functions of angles greater than 90° (or less than 0°) can be found using corresponding acuteangles called reference angles.

  5. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 0 90° < < 180°; π π << 0 2 0 ' ' ' 0 0 0 – 0 0 = 180° Degrees: Radians:= π –

  6. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 180° < < 270°; 3π π << 2 0 0 0 ' ' ' 0 0 0 – = 0 0 180° Degrees: π – Radians:=

  7. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 270° < < 360°; 3π 2π << 2 0 0 0 ' ' ' 0 0 0 – = 0 0 360° Degrees: 2π – Radians:=

  8. Finding Reference Angles Find the reference angle for each angle . 0 0 0 0 0 5π = 320° = – 6 ' ' ' 0 0 0 Because 270°< < 360°, the reference angle is = 360° – 320° = 40°. Because is coterminal with and π < < , the reference angle is = – π = . 7π 7π 3π 6 6 2 7π π 6 6 SOLUTION

  9. Evaluating Trigonometric Functions Given a Point CONCEPT EVALUATING TRIGONOMETRIC FUNCTIONS SUMMARY Use these steps to evaluate a trigonometric function of any angle . 0 1 3 2 Find the reference angle . 0 0 ' ' 0 0 Evaluate the trigonometric function for angle . Use the quadrant in which lies to determine thesign of the trigonometric function value of .

  10. Evaluating Trigonometric Functions Given a Point CONCEPT EVALUATING TRIGONOMETRIC FUNCTIONS SUMMARY Quadrant II Quadrant I sin , csc : + sin , csc : + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cos , sec : – cos , sec : + tan , cot :– tan , cot :+ Quadrant III Quadrant IV sin , csc : – sin , csc : – cos , sec : + cos , sec : – tan , cot :– tan , cot :+ Signs of Function Values

  11. Using Reference Angles to Evaluate Trigonometric Functions =30° =–210° 0 ' 0 ' 0 The reference angle is = 180° – 150° = 30°. 3 tan (– 210°) = – tan 30° = – 3 Evaluate tan (– 210°). SOLUTION The angle –210°is coterminal with 150°. The tangent function is negative in Quadrant II, so you can write:

  12. Using Reference Angles to Evaluate Trigonometric Functions Evaluate csc. π = 4 = 0 11π 11π 3π The angle is coterminal with . ' 0 4 4 4 ' 0 π 3π The reference angle is = π – = . 4 4 11π 11π π 4 csc = csc = 2 4 4 SOLUTION The cosecant function is positive in Quadrant II, so you can write:

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