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T RIGONOMETRIC F UNCTIONS OF A NY A NGLE

Learn how to define and evaluate the six trigonometric functions of any angle using the general definition. Understand the concept of reference angles and how they can be used to find the values of trigonometric functions in different quadrants.

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T RIGONOMETRIC F UNCTIONS OF A NY A NGLE

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  1. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS 0 0 0 0 Let be an angle in standard position and (x, y) be any point (except the origin) on the terminal side of . The six trigonometric functions of are defined as follows. (x, y) r

  2. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS r csc = , y 0 sin = 0 0 y y r r y 0 y r

  3. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS r sec = , x 0 0 x x cos = 0 r r x 0 x r

  4. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS y x tan = , x 0 cot = , y 0 0 0 x y x y 0 y x

  5. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS (x, y) 0 r r = x2+y2. Pythagorean theorem gives

  6. 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

  7. Evaluating Trigonometric Functions Given a Point r 0 y 5 r 4 csc = = – 0 sin = = – 0 y r 4 5 3 x r 5 0 cos = = 0 sec = = r 5 x 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)

  8. 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.

  9. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 90 < < 180; 0   << 0 2 0 ' ' ' 0 0 0 – = 180 0 0 Degrees: Radians:=  –

  10. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 180 < < 270; 3  << 2 0 0 0 ' ' ' 0 0 0 – = 180 0 0 Degrees:  – Radians:=

  11. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 270 < < 360; 3 2 << 2 0 0 0 ' ' ' 0 0 0 – = 360 0 0 Degrees: 2 – Radians:=

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

  13. 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 .

  14. 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

  15. 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:

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

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