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Double Angle Identities

Simplifying trigonometric functions with twice a given angle.

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Double, Half, and Power Reducing Identities

These identities are significantly more involved and less intuitive than previous identities.  By practicing and working with these advanced identities, your toolbox and fluency substituting and proving on your own will increase.  Each identity in this concept is named aptly.  Double angles work on finding  if you already know .  Half angles allow you to find  if you already know .  Power reducing identities allow you to find  if you know the sine and cosine of

What is

Double Angle, Half Angle, and Power Reducing Identities

Double Angle Identities

The double angle identities are proved by applying the sum and difference identities.  They are left as review problems.  These are the double angle identities.

Half Angle Identities

The half angle identities are a rewritten version of the power reducing identities.  The proofs are left as review problems.

Power Reducing Identities

The power reducing identities allow you to write a trigonometric function that is squared in terms of smaller powers.  The proofs are left as examples and review problems. 

Power reducing identities are most useful when you are asked to rewrite expressions such as  as an expression without powers greater than one. While  does technically simplify this expression as necessary, you should try to get the terms to sum together not multiply together.

Examples

Example 1

Earlier, you were asked to find . In order to fully identify  you need to use the power reducing formula.

Example 2

Write the following expression with only  and : .

Example 3

Use half angles to find an exact value of  without using a calculator. 

Sometimes you may be requested to get all the radicals out of the denominator.

Example 4

Prove the power reducing identity for sine.

Start with the double angle identity for cosine.

This expression is an equivalent expression to the double angle identity and is often considered an alternate form.

Example 5

Simplify the following identity: .

Here are the steps:

Review

Prove the following identities.

1. 

2.

3.

4. 

5. 

6. 

7. 

8. 

9. 

10. 

Find the value of each expression using half angle identities.

11. 

12. 

13. 

14.  Show that .

15.  Show that .

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.4. 

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Vocabulary

TermDefinition
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.
identity An identity is a mathematical sentence involving the symbol “=” that is always true for variables within the domains of the expressions on either side.
power reducing identity A power reducing identity relates the power of a trigonometric function containing a given argument to a set of trigonometric functions, each containing the original argument.

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