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13.3 Evaluate Trigonometric Functions of Any Angle Goal Evaluate trigonometric functions of any angle. Your Notes VOCABULARY Unit circle The circle x2 + y2 = 1, which has center (0, 0) and radius 1 Quadrantal angle An angle in standard position whose terminal side lies on an axis Reference angle The reference angle for is the acute angle ’ formed by the terminal side of and the x-axis. GENERAL DEFINITIONS OF TRIGONOMETRIC FUNCTIONS Let be an angle in standard position, and let (x, y) be the point where the terminal side of intersects the circle x2 + y2 = r2. The six trigonometric functions of are defined as follows: sin = y r csc = r ,y 0 y cos = x r sec = r ,x 0 x tan = y ,x 0 x cot = x ,y 0 y Your Notes Example 1 Evaluate trigonometric functions given a point Let (12, 5) be a point on the terminal side of an angle in standard position. Evaluate the six trigonometric functions of . Use the Pythagorean theorem to find the value of r. r = x2 y2 = 2 ( 12 ) 2 5 = 169 = _13_ Using x = 12, y = 5, and r = _13_ , you can write: sin = 5 y = 13 r csc = cos = x r = 12 13 sec = 13 r = 12 x tan = y = 5 cot = x y x 12 r y = 13 5 = 12 5 Checkpoint Complete the following exercise. 1. Let (6, 8) be a point on the terminal side of an angle in standard position. Evaluate the six trigonometric functions of . sin = sec = 4 3 4 5 , cos = ,tan = , csc = 5 5 3 4 5 3 , cot = 3 4 , THE UNIT CIRCLE The circle x2 + y2 = 1, which has center (0, 0) and radius 1, is called the unit circle. sin = y y = =_y_ r 1 cos = x x = =_x_ r 1 Your Notes Example 2 Use the unit circle Use the unit circle to evaluate the six trigonometric functions of = 450°. Draw the unit circle, then draw the angle = 450° in standard position. The terminal side of intersects the unit circle at (_0_, _1_), so use x = _0_ and y = _1_ to evaluate the trigonometric functions. sin = y 1 = =_1_ 1 r csc = cos = x 0 = =_0_ r 1 sec = tan = y cot = x = 1 _undefined_ 0 r 1 = =_1_ y 1 r x = 1 _undefined_ 0 x 0 = =_0_ y 1 Checkpoint Complete the following exercise. 2. Use the unit circle to evaluate the six trigonometric functions of = 360°. sin = 0, cos = 1, tan = 0, csc is undefined, sec = 1, cot is undefined. Your Notes REFERENCE ANGLE RELATIONSHIPS Let be an angle in standard position. The reference angle for is the acute angle ’ formed by the terminal side of and the x-axis. The relationship between and ’ is shown below for nonquadrantal angles such that 90° < < 360°( 2 < < 2). Quadrant II Quadrant III Quadrant IV Degrees: ’ =180° ’ = 180° ’ =360° Radians: ’ = ’ = – ’ =2 Example 3 Find reference angles Find the reference angle ’ for (a) = 165° and 7 (b) = . 4 Solution a. Note that is coterminal with _195°_, whose terminal side lies in Quadrant _III_. So, ’ = _195°_ _180°_ = _15°_ . b. The terminal side of lies in Quadrant _IV_ . So, ’ = _2_ 7 = . 4 4 Your Notes EVALUATING TRIGONOMETRIC FUNCTIONS Use these steps to evaluate a trigonometric function for any angle : STEP 1 Find the reference angle _’_. STEP 2 Evaluate the trigonometric functions for _’_. STEP 3 Determine the sign of the trigonometric function value from the quadrant in which __ lies. Example 4 Use reference angles to evaluate functions Evaluate (a) cos(225° ) and (b) cot 10 . 3 a. The angle 225° is coterminal with _135°_. The reference angle is ’ = _180°_ _135°_ = _45°_ . The cosine function is negative in Quadrant _II_, so you can write: 2 cos(225°) = cos( _45°_) = 2 4 b. The angle 10 is coterminal with . The reference 3 3 angle is ' = 4 __= . The cotangent 3 3 function is positive in Quadrant _III_, so you can write: cot ( 10 ) =cot 3 3 = 3 3 Checkpoint Complete the following exercises. 3. Find the reference angle for = 150°. 30° 4. Evaluate sec 300°. 2 Homework ________________________________________________________________________ ________________________________________________________________________