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Lesson 4.4 Trigonometric Functions of Any Angle Trigonometric Functions of Any Angle Definitions of Trigonometric Functions of Any Angle: Let be an angle in standard position with (x, y) a point on the Terminal side of and r x 2 y 2 0 y sin r x cos r r csc , y 0 y r sec , x 0 x x y tan , x 0 cot , x 0 y x r Trigonometric Functions of Any Angle Example 1: Let (8, - 6) be a point on the terminal side of . Find the sine, cosine, and tangent of . Solution: 8 6 r 2 2 Step 1: Find r. 2 64 36 r 2 100 r 2 10 r Step 2: Apply the definitions for sine, cosine, and tangent. x 8 4 cos r 10 5 y 6 3 tan x 8 4 y 6 3 sin r 10 5 Trigonometric Functions of Any Angle The signs of the trigonometric functions in the four quadrants can be easily determined by applying CAST. CAST let’s one know where the trigonometric functions are positive. y S Sine & Cosecant A All trig functions are positive. are positive. T x C Tangent & Cotangent Cosine & Secant are positive. are positive. Remember the acronym: All Students Take Calculus Trigonometric Functions of Any Angle Example 2: Given sin 11 and tan 0, find the value of the remaining trig functions. 61 Step 1: Determine the quadrant that the terminal side of lies. Sine is positive in Quad I and Quad II, while tangent is positive in Quad I and Quad III. Therefore, the terminal side must lie in Quad I. Step 2: Determine the value of r using the given value of sine. r x2 y 2 61 x 2 112 61 2 x 2 121 3721 x 2 121 3600 x 2 60 x 2 Step 3: State the values for the remaining trig functions by applying the definitions. 60 61 cos sec 61 60 11 60 tan cot 60 11 61 csc 11 Trigonometric Functions of Any Angle The values of trigonometric functions of angles greater than 90 can be determined by using a reference angle. Definition of a reference angle: Let be an angle in standard position. Its reference angle is the acute positive angle ′ formed by the terminal side of and the nearest x-axis. ′ ′ In Quad II In Quad III ′ In Quad IV 2 Trigonometric Functions of Any Angle Example 3: Find the reference angle for 7 9 Step 1: Determine the quadrant that terminal side lies. The terminal side for this angle lies in Quad II. Step 2: Determine the value of the nearest x-axis. The nearest x-axis holds a value of . Step 3: Calculate the value for the reference angle. Remember the reference angle must be an acute angle and positive. 2 9 7 9 Trigonometric Functions of Any Angle Example 4: Find the exact values of the six trigonometric functions for 10 3 First, sketch the angle and determine the angle’s simplest positive coterminal angle. 10 4 2 3 3 Second, determine the new angle’s reference angle based on where the terminal side lies. 4 3 3 ′ Third, give the trigonometric values for the original angle based on the quadrant the terminal side is located and the reference angle. 10 3 10 2 3 3 2 10 1 cos 3 2 sin tan 10 3 3 3 3 10 sec 2 3 csc cot 10 3 3 3 Trigonometric Functions of Any Angle Try these: 15 17 8 cos 17 15 tan 8 sin 1. Determine the exact values of the six trigonometric functions of the angle given (- 8, - 15) lies on the terminal side. 2. Find the values of the six trigonometric functions of giventan = - 4/3 and sin < 0. sin cos 3 5 tan 3. Find the reference angle for: 197 180 17 a. 197 12 2 2 b. 12/7 c. - 3.68 7 7 4 5 17 15 17 sec 8 8 cot 15 csc csc sec 4 3 5 4 5 3 cot 3 4 3.68 6.28 3.00 3.14 3.00 0.14 Trigonometric Functions of Any Angle What you should know: 1. How to evaluate the trigonometric functions of any angle. 2. How find and use the reference angle to evaluate trigonometric functions.