13.3 Evaluating Trigonometric Functions

terminal side of an angle in standard position. Evaluate the 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 . r (3, – 4) SOLUTION Use the Pythagorean theorem to find the value of r. r = x 2 + y 2 = 3 2 + (– 4 ) 2 = 25 = 5

you can write the following: Evaluating Trigonometric Functions Given a Point r Using x = 3, y = – 4, and r = 5, you can write the following: (3, – 4) 4 5 sin = = – y r csc = = – r y 5 4 cos = = x r 3 5 sec = = r x 5 3 tan = = – y x 4 3 cot = = – x 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 be an angle in standard position. Its reference angle is the acute angle (read theta prime) formed by the terminal side of and the x-axis. '

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

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

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

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

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

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

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

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