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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.