Topic: Trigonometric Identities

Subtopic: Half-Angle Identities

**Overview**

Continuing to expand our arsenal of useful trigonometric formulae, in today's lesson we study the __half-angle identities__ which expand trig functions of a half-angle such as `cos(x/2)`.

Caution: To avoid a common error, when determining the sign of the answer from a half-angle formula, remember that it is based on the quadrant of the half-angle, not the quadrant of the whole angle.

The half-angle identities are the last of the identities we study. In my opinion, you do not need to memorize all the trig identities we studied. However you MUST memorize the Pythagorean identities and the double-angle identities since they are used so frequently here and in calculus and engineering. The rest though can become a bit overwhelming. For them you may want to use a "cheat sheet" of identities (see below).

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 5.4.1 Use half-angle formulae to evaluate trig functions
- 5.4.2 Use half-angle formulae to prove other trig identities
- 5.4.3 Solve problems involving a half-angle

**Terminology**

Terms you should be able to define: half-angle

You do __not__ need to memorize these **HALF ANGLE IDENTITIES**.

You do need to know how to apply them. Keep a copy handy.

**Mini-Lectures and Examples**

STUDY: Trigonometric Identities - Double-Angle IDs & Half-Angle IDs

**Supplementary Resources (required reading!)**

This is also an appropriate time to start using a reference sheet of identities.

**Download/Print:**

Prof. Keely's Trigonometric Identities and Formulas Sheet

You may refer to this sheet during a test! You do NOT need to memorize these formulas.

- Other formula sheets you might want to use for your studies include Trigonometry Review Sheet from Cengage or Trigonometry Cheat Sheet from Prof. Paul Dawkins, Lamar Univ.. This latter sheet provides a particularly nice summary of formulae.

You may NOT refer to these two sheets or anything similar to them during a test! These sheets contain information the Clark Mathematics Department does NOT allow for use during a test such as the unit circle.

You must MEMORIZE the unit circle, reciprocal identities, and ratio identities and NOT refer to notes for these items during a test! I recommend you also memorize the Pythagorean identities. but this is not required by the Maths Dept.

- To help you memorize the reciprocal and ratio identities try this little game: Trig Formulas Game from PurposeGames.com.

**Supplementary Resources (optional)**

Mini-Lesson: __Half Angle Identities__

Mini-Lesson: __Verifying Identities: Sum, Difference, Double, and Half Angle Identities__

__Example: Rewrite a Trig Expression Using a Half Angle Identity__

__Example: Determine a Cosine Function Value Using a Half Angle Identity__

__Example: Determine a Sine Function Value Using a Half Angle Identity__

__Example: Determine a Tangent Function Value Using a Half Angle Identity__

rev. 2020-10-19