Now that we’ve learned the fundamental identities and how to use them to prove/verify identities, it’s now time to spread our wings and learn some other very useful formulas: Sum and Difference Identities.
While the Sum and Difference Identities are derived from Euler’s Formula dealing with complex numbers and the rules of exponents, as Brown Math nicely shows, we will be focusing on their application and use, because these little formulas can do some amazing things!
First, we will investigate how to use these Sum and Difference Identities to evaluate angles that aren’t on our Unit Circle.
For example, imagine we want to find the exact value of sin(15). What we will quickly see is that we can create 15 degrees by saying 60-45, which are two known values on our unit circle.
Sum Identity for Sine
Along, with our Sum and Difference Formulas we will quickly be able to arrive at our answer without the use of a calculator by saying sin(15) = sin(60-45)! Cool!
Now, some textbooks will refer to the Sum and Difference Identities or Formulas as the Trigonometric Angle Addition and Subtraction Identities.
Do not be alarmed, because they all mean the same thing. Phew!
Again, what’s so great about these special identities is that the helps us to find the trig values of sums/differences of angles in terms of other trig values, as Khan Academy discusses.
Lastly, we will learn how to verify some tricky identities using our skills and steps from our previous lessons as well as applying our new Sum and Difference Identities.
How to Solve Sum and Difference Identities – Video
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