Lesson 4.4 Trigonometric Functions of Any Angle

Let  be an angle in standard position with (x, y) a point on the Terminal side of  and Trigonometric Functions of Any Angle Definitions of Trigonometric Functions of Any Angle: 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: Step 1: Find r. Step 2: Apply the definitions for sine, cosine, and tangent.

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. x y A S T A ll trig functions are positive. S ine & Cosecant are positive. T angent & Cotangent are positive. Remember the acronym: A ll S tudents T ake C alculus C C osine & Secant are positive.

Example 2: Given, find the value of the remaining trig functions. Trigonometric Functions of Any Angle 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. Step 3: State the values for the remaining trig functions by applying the definitions.

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   ′ ′

Trigonometric Functions of Any Angle Example 3: Find the reference angle for 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.

Trigonometric Functions of Any Angle Example 4: Find the exact values of the six trigonometric functions for First, sketch the angle and determine the angle’s simplest positive coterminal angle.   ′ ′ Second, determine the new angle’s reference angle based on where the terminal side lies. Third, give the trigonometric values for the original angle based on the quadrant the terminal side is located and the reference angle.

Trigonometric Functions of Any Angle Try these: 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  < Find the reference angle for: a. 197  b. 12  /7 c

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.