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.

2log 6 x2 6x

6.8 NotesYou 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:

8sin

11A

6.8 NotesTo 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.”

1 8sin

11A

8sin

11A

6.8 NotesAnother 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.”

8sin

11A arc

8sin

11A

6.8 NotesExample 1:

Find the inverse relation equation of .

“Beta is the inverse cosine of x.”

1cos x

cosx

6.8 NotesExample 2:

Find the inverse relation equation of .

“Theta is the arc-tangent of 1.”

arctan1

tan 1

6.8 NotesExample 3:

Find the inverse relation equation of

“Alpha is the inverse sine of negative square root of two divided by two.”

1 2sin

2

2sin

2

6.8 Notes – practice problems:1.

2.

3. 3 tan

sinx

1tan 3 tan 3or arc

1sin sinx or arc x

1 2 2cos arccos

3 3or

2cos

3

6.8 NotesThe second thing you will learn to do in this lesson is to solve inverse trigonometric relation equations such as

and .

tan 3arc

1 2cos

3

6.8 NotesFrom 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, .1 3tan

3

3tan

3

6.8 NotesTherefore, 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 .1 3tan

3x

0 360x

30 ,210x

6.8 NotesExample 2:

Find the values of x in the interval

that satisfy the equation . 1sin 1x

0 2x

3

2x

6.8 NotesExample 3:

Find the values of x in the interval

that satisfy the equation .

0 360x

45 ,315x

1sec 2x

1 2cos

2x

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

1. 2.

Find values of x in the interval .

3. 4.

1 3sin

2x

0 360x

60 ,120x 135 ,315x

arctan 1x

0 2x

2 4,

3 3x

1 3cot

3x

4,

3 3x

1 1cos

2x