<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

3.12: Trigonometric Equations Using Half Angle Formulas

Difficulty Level: At Grade Created by: CK-12
Atoms Practice
Estimated5 minsto complete
%
Progress
Practice Trigonometric Equations Using Half Angle Formulas
 
 
 
MEMORY METER
This indicates how strong in your memory this concept is
Practice
Progress
Estimated5 minsto complete
%
Estimated5 minsto complete
%
Practice Now
MEMORY METER
This indicates how strong in your memory this concept is
Turn In

As you've seen many times, the ability to find the values of trig functions for a variety of angles is a critical component to a course in Trigonometry. If you were given an angle as the argument of a trig function that was half of an angle you were familiar with, could you solve the trig function?

For example, if you were asked to find

would you be able to do it? Keep reading, and in this section you'll learn how to do this.

Using Half Angle Formulas on Trigonometric Equations

It is easy to remember the values of trigonometric functions for certain common values of . However, sometimes there will be fractional values of known trig functions, such as wanting to know the sine of half of the angle that you are familiar with. In situations like that, a half angle identity can prove valuable to help compute the value of the trig function.

 

 

 In addition, half angle identities can be used to simplify problems to solve for certain angles that satisfy an expression. To do this, first remember the half angle identities for sine and cosine:

if is located in either the first or second quadrant.

if is located in the third or fourth quadrant.

if is located in either the first or fourth quadrant.

if is located in either the second or fourth quadrant.

When attempting to solve equations using a half angle identity, look for a place to substitute using one of the above identities. This can help simplify the equation to be solved.

Let's look at some problems that use the half angle formula. 

1. Solve the trigonometric equation over the interval .

Then or , which is .

.

2. Solve for

To solve , first we need to isolate cosine, then use the half angle formula.

when

3. Solve for

To solve , first isolate tangent, then use the half angle formula.

Using your graphing calculator, when

Examples

Example 1

Earlier, you were asked to solve sin 22.5°.

Knowing the half angle formulas, you can compute easily:

Example 2

Find the exact value of

Example 3

Find the exact value of

Example 4

Find the exact value of

Review

Use half angle identities to find the exact value of each expression.

Use half angle identities to help solve each of the following equations on the interval .

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.12. 

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

Vocabulary

Half Angle Identity

A half angle identity relates a trigonometric function of one half of an argument to a set of trigonometric functions, each containing the original argument.

Image Attributions

Show Hide Details
Description
Difficulty Level:
At Grade
Subjects:
Grades:
Date Created:
Sep 26, 2012
Last Modified:
Mar 23, 2016
Save or share your relevant files like activites, homework and worksheet.
To add resources, you must be the owner of the Modality. Click Customize to make your own copy.
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.TRG.356.L.1
Here