6.8 Notes In this lesson you will learn how to… write trigonometric equations as inverse trigonometric relation equations. find the values that satisfy an inverse trigonometric relation equation.

6.8 Notes Exponential and logarithmic functions are inverses of each other. From today’s “do now,” the equivalent logarithmic form of the exponential equation,, is. In this lesson, you will learn to write the inverse trigonometric relation equation given a trigonometric function equation.

6.8 Notes You have used inverse trigonometric functions to find angle measurements of triangles. To find the measure of angle A in the triangle pictured below you set up the trigonometric equation shown:

6.8 Notes To solve for A, take the inverse sine of both sides of the equation: The bottom equation is the inverse relation equation of the top equation. It is read, “A is the inverse sine of eight elevenths.”

6.8 Notes Another notation for the inverse of a trigonometric function is to use the prefix arc- with the trigonometric function’s abbreviation instead of the -1 exponent on the trigonometric function’s abbreviation. This is read, “A is the arc-sine of eight elevenths.”

6.8 Notes Example 1: Find the inverse relation equation of. “Beta is the inverse cosine of x.”

6.8 Notes Example 2: Find the inverse relation equation of. “Theta is the arc-tangent of 1.”

6.8 Notes Example 3: Find the inverse relation equation of “Alpha is the inverse sine of negative square root of two divided by two.”

6.8 Notes – practice problems:

6.8 Notes The second thing you will learn to do in this lesson is to solve inverse trigonometric relation equations such as and.

6.8 Notes From today’s “do now,” due to the periodic nature of the trigonometric functions, there are an infinite number of values of θ that satisfy the equation,. Therefore, there are an infinite of solutions to its corresponding inverse trigonometric relation equation,.

6.8 Notes Therefore, instructions for solving these equations will include an interval, typically the first positive revolution of the unit circle. Example 1: Find the values of x in the interval that satisfy the equation.

6.8 Notes Example 2: Find the values of x in the interval that satisfy the equation.

6.8 Notes Example 3: Find the values of x in the interval that satisfy the equation.

6.8 Notes – practice problems: Find values of x in the interval Find values of x in the interval. 3.4.