Pre-Calculus 12

Solving Trigonometric Equations, Identities and Proofs

Special Angles:

Unit Circle:

Note: ( r = radius )

Radians: The angle created when the central angle has an arc

length equal to the radius. Conversion factor

( angles must be in radians. )

Solving equations

Find the solution for that is over a finite interval and also a general solution.

Ex 1. Solve , given and general solution.

: the angle is in quadrant III or IV. The

the general solution is

Ex 2. Solve , general solutions.

: the angle is in all 4 quadrants. The

These would all be within one rotation. Notice that separates the angles in quadrant I and III … quadrant II and IV. The general solution for this would be….

Ex 3. Solve , general solution.

For

For let so Write a general solution for

so this is simplified to

Change back to

What if Ex 3. was to solve over an interval of ?

for the sine equation and…

for the tangent equation.

This means there are 9 solutions for this equation.

Ex 4. Solve , for You need to use an identity here

The above examples show various levels of difficulty involved with solving equations. There are quadratic trigonometric equations, trigonometric equations with , and trigonometric equations that involve identities.

Identities and Proofs

Use your identity sheet to determine if you are correctly using the identities appropriately. In some questions you will be asked to simplify an expression or determine an appropriate identity.

Ex 5. Simplify

Restrictions: In the above example we can say ; however, this is not true for all values.

How to do a Proof ? Unfortunately, there is no straight forward answer. You can not move terms from one side to the other. You can only manipulate one side at a time until both sides are equal. Here are a few suggestions.

1. Pick one side and convert to sine or cosine. ***

2. Look for obvious identities : Pythagorean

3. Look for factoring :

4. Try something else. If you are getting nowhere, try a different approach.