TRIGONOMETRIC FUNCTIONS OF ANY ANGLE GENERAL DEFINITION OF

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

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS y csc sin 00 == r r , y 0 y yr y r 0

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS x sec 0 == r , x 0 cos rx x r 0

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS yx tan cot 00== , x, y 0 0 xy x y y x 0

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE GENERAL DEFINITION OF TRIGNONOMETRIC FUNCTIONS (x, y) 0 r Pythagorean theorem gives r = x 2 +y 2.

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

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

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

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

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

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

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

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

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

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

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