Homework Quiz 4.3 A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°. How tall is the flagpole? To receive credit you must have a diagram completely labeled, the trig function that you used to solve, and the answer. 4.4 Trigonometric Functions of Any Angle Precalculus

Example: In traveling across flat land you notice a mountain directly in front of you. Its angle of elevation to the peak is 3.5 degrees. After you drive 13 miles closer to the mountain, the angle of elevation is 9 degrees. Approximate the height of the mountain. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Ex: At a certain distance, the angle of elevation to the top of a tree is 60 degrees. From 40 feet further back the angle of elevation is 45 degrees. Find the height of the building. To receive credit you must have a diagram completely labeled, the trig function that you used to solve, and the answer. 4.4 Trigonometric Functions of Any Angle Precalculus

4.4 Trig Functions of Any Angle 2014 Objectives: Evaluate trigonometric functions of any angle. Use reference angles to evaluate trig functions.

Trig Functions of Any Angle Given an angle  in standard position with (x,y) a point on the terminal side of  y (x,y)  r x 4.4 Trigonometric Functions of Any Angle Precalculus

Example 1 Let (-3,4) be a point on the terminal side of . Find the sine, cosine and tangent of . 4.4 Trigonometric Functions of Any Angle Precalculus

How do I know the sign of the trig function? One way to think of the sign of a function is to remember the variable it is defined by. Since the radius is always positive, only the signs of x and y influence the sign of the function. In any given quadrant: cosine and secant have the same sign as x sine and cosecant have the same sign as y tangent and cotangent have the same sign as the ratio of x and y 4.4 Trigonometric Functions of Any Angle Precalculus

Determine the Quadrant in Which the Angle Lies 1) 2) 3) 4) 4.4 Trigonometric Functions of Any Angle Precalculus

Example 2 Given Find the values of the six trig functions of  4.4 Trigonometric Functions of Any Angle Precalculus

Example 3 Given Find the values of the six trig functions of  4.4 Trigonometric Functions of Any Angle Precalculus

Reference Angles Let  be an angle in standard position. Its reference angle is the acute angle ’ formed by the terminal side of  and the x-axis.  ’  ’  ’ 4.4 Trigonometric Functions of Any Angle Precalculus

Calculating Reference Angles Quadrant II Quadrant III Quadrant IV  ’  ’  ’ 4.4 Trigonometric Functions of Any Angle Precalculus

Example 4 Find the reference angle ’ 1)  = 300° 2)  = 3)  = -135° 4.4 Trigonometric Functions of Any Angle Precalculus

Evaluating Trig Functions of Any Angle To find the value of a trig function of any angle : Evaluate the function for the associated reference angle ’. Determine the sign of the trig function based on the quadrant in which  lies. 4.4 Trigonometric Functions of Any Angle Precalculus

Example 4 Use reference angles to evaluate the trig function 1) 2) 3) 4.4 Trigonometric Functions of Any Angle Precalculus

Example 5 – Using Trig Identities Let  be an angle in Quadrant II such that sin =1/3. Find cos  and tan . 4.4 Trigonometric Functions of Any Angle Precalculus

Closure Explain how to use reference angles to determine trig functions of any angle 4.4 Trigonometric Functions of Any Angle Precalculus

Homework 4.4 pg 284 1-33 EOO, 53-73 odd, 91,93 4.4 Trigonometric Functions of Any Angle Precalculus

Bellwork Find sin  Determine the sign (positive or negative) of the six trig functions for angles in Quadrant I, Quadrant II, Quadrant III, and Quadrant IV 4 7  4.4 Trigonometric Functions of Any Angle Precalculus