Sum & Difference Identities Game

Sum & Difference Identities Quiz/Game
The Sum and Difference Formulas game is all about recognizing how to “expand” or “collapse” trigonometric expressions. These formulas are the “blueprints” for combining two different angles (A and B). Scroll down the page for a more detailed explanation.
 

 

How to Play

1. Identify the Function First:
   Look at the functions and look at the signs.
   
2. Recall the Rule:
   Mentally scan your “identity bank.”
   
3. Select the Match:
   Click one of the four buttons.
   Green Flash: You matched correctly
   Red Flash: You missed it. The game will immediately highlight the correct answer in green so you can memorize it for the next round.
   
4. Scoring and Content
   The game tracks your progress in the top-left corner (Score: Correct / Total Attempts).
    
   

How to remember the Sum & Difference Identities
The Sine Identities:
Sine is often described as “friendly” and “cooperative” because it mixes the functions together and keeps the original sign.
Sum: sin(A + B) = sin A cos B + cos A sin B
Difference: sin(A - B) = sin A cos B - cos A sin B
Key Pattern:
Mixing: It alternates sin and cos.
Sign Stability: If the problem has a +, the expansion has a +.

The Cosine Identities:
Cosine is the “selfish rebel.” It groups its own terms together and always flips the sign of the operation.
Sum: cos(A + B) = cos A cos B - sin A sin B
Difference: cos(A - B) = cos A cos B + sin A sin B
Key Pattern:
Grouping: It keeps cos terms together and sin terms together.
Sign Flip: If you are adding the angles, you subtract the terms.

The Tangent Identities:
Since \(\tan = \frac{\sin}{\cos}\), its identity is a fraction. It follows a “Top follows, Bottom rebels” rule regarding the plus/minus signs.
Sum: \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\)
Difference: \(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\)
 

Why Use Them?
These identities are essential for several reasons:
Finding Exact Values: You can find the exact radical form of sin(15°) by using 45° - 30°.
Simplifying Expressions: In physics and engineering, you often encounter waves that interfere with each other; these formulas allow you to combine multiple wave equations into one.
Deriving Other Rules: They can be used to derive the Double-Angle identities. If you let B = A, then sin(A + A) becomes the formula for sin(2A).
 

Example Walkthrough: cos(75°)
To solve cos(75°), we rewrite it as cos(45° + 30°).
Use the Cosine Sum formula: cos A cos B - sin A sin B
Plug in values: cos(45°)cos(30°) - sin(45°)\sin(30°)
Substitute from the unit circle: \((\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) - (\frac{\sqrt{2}}{2})(\frac{1}{2})\)
Result: \(\frac{\sqrt{6} - \sqrt{2}}{4}\)
 

Have a look at this lesson on Sum and Difference
Sum & Difference Identities
 

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