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T RIGONOMETRIC F UNCTIONS OF A NY A NGLE. 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.
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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
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
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
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
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS (x, y) 0 r r = x2+y2. Pythagorean theorem gives
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
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)
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.
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 90 < < 180; 0 << 0 2 0 ' ' ' 0 0 0 – = 180 0 0 Degrees: Radians:= –
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 180 < < 270; 3 << 2 0 0 0 ' ' ' 0 0 0 – = 180 0 0 Degrees: – Radians:=
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 270 < < 360; 3 2 << 2 0 0 0 ' ' ' 0 0 0 – = 360 0 0 Degrees: 2 – Radians:=
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
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 .
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
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:
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: