Trigonometric Functions of Any Angle Section 9 3

What You Will Learn Evaluate trigonometric functions of any angle. Find and use reference

Example: Evaluating Trigonometric Functions Given a Point Let (− 4, 3) be a point

Take Note It is convenient to use the unit circle to find trigonometric functions

Using the Unit Circle Use the unit circle to evaluate the six trigonometric functions

Reference angle Let θ be an angle in standard position. Its reference angle is

Reference angles are always measured between the x axis and the terminal side

Finding the Trig functions of any angle 1. Find the reference angle 2. Determine

Drawing Reference Angles Find the reference angle θ'�, and sketch θ and θ' in

Using Reference Angles to Evaluate Functions Evaluate (a) tan(− 240º) and (b) csc 17π/6.

Now: Find all 6 trig functions of an angle Find the six trigonometric functions

Now, let’s make it tougher The terminal side of θ lies on the line

Try These: Find the values of the six trig functions of θ with the

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Trigonometric Functions of Any Angle Section 9. 3

What You Will Learn Evaluate trigonometric functions of any angle. Find and use reference angles to evaluate trigonometric functions

Take Note

Example: Evaluating Trigonometric Functions Given a Point Let (− 4, 3) be a point on the terminal side of an angle θ in standard position. Evaluate the six trigonometric functions of θ.

Take Note It is convenient to use the unit circle to find trigonometric functions of quadrantal angles. A quadrantal angle is an angle in standard position whose terminal side lies on an axis. The measure of a quadrantal angle is always a multiple of 90º, or π/2 radians.

Using the Unit Circle Use the unit circle to evaluate the six trigonometric functions of θ = 270º. Find the values of the six trigonometric functions. Let x = 0 and y = − 1 to evaluate the trigonometric functions.

Take Note

Reference angle Let θ be an angle in standard position. Its reference angle is the acute angle θ‘ formed by the terminal side of θ and the horizontal axis. 9

Take Note

Reference angles are always measured between the x axis and the terminal side of the angle (always +ves !) Notice the butterfly shape 11

Finding the Trig functions of any angle 1. Find the reference angle 2. Determine the value of the given trig function of the reference angle. 3. Determine the sign—based on the quadrant of the given (original) angle 12

Drawing Reference Angles Find the reference angle θ'�, and sketch θ and θ' in standard position. 1) θ = -145° 2) 13

Using Reference Angles to Evaluate Functions Evaluate (a) tan(− 240º) and (b) csc 17π/6.

Now: Find all 6 trig functions of an angle Find the six trigonometric functions of θ with the given constraint. 15

Now, let’s make it tougher The terminal side of θ lies on the line y = (1/3)x in quadrant I. Find the values of the 6 trig functions of θ by finding a point on the line. sin θ = cos θ = csc θ = sec θ = tan θ = cot θ = 16

Try These: Find the values of the six trig functions of θ with the given constraints. sin θ = 0; where sec θ = -1 Find the reference angle θ‘ and sketch it and the angle θ in standard position. a) θ = 3. 5 b) θ = 750° 17