Apply Sum & Difference Identities Game

Apply Sum & Difference Identities Quiz/Game
This game is a study tool designed to help you internalize the Sum and Difference Identities for Sine, Cosine, and Tangent. It focuses on the specific “tricky” angles like 15° and 75° that aren’t usually on a basic unit circle. Scroll down the page for a more detailed explanation.
 

 

How to Play

1. Choose Your Challenge
   When you start, you must choose one of two modes. These represent the two different steps in solving a complex trig problem:
   Mode A: Identity Expansion
   This mode tests your memory of the formulas. You see a problem like cos(75°) and must identify the expanded identity cos A cos B - sin A sin B. Strategy: Look for the “Sine Mix” vs. the “Cosine Group."
   The “Cosine Flip”: Remember that for Cosine, the sign inside the expansion is the opposite of the angle’s math ($75^\circ = 45+30$, so the expansion uses a minus sign).
   Mode B: Calculated Result
   This mode skips the formula and asks for the final numerical value. It tests your ability to visualize the unit circle and combine the fractions.
   

2. Gameplay
   The Problem: A random trigonometric expression appears in the center (e.g., sin(15°)).
   Multiple Choice: Four buttons appear below. Only one is mathematically correct for your chosen mode.
   Instant Feedback:
   Green: You nailed it.
   Red: You missed it, but the game will immediately highlight the correct answer in green so you can learn from the mistake.
   Progression: Your score is tracked at the top. Click “Next Question” to keep practicing
    
   

How to Apply Sum & Difference Identities

1. The Formulas
   The identities act as the “instruction manual” for how to combine two angles (A and B).
   Function: Sine
   Identity: \(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\)
   The “Memory Trick”: “The Mixer”: Sine mixes Sine and Cosine together. The sign stays the same.
   
   Function: Cosine
   Identity:: \(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\)
   The “Memory Trick”: “The Loner”: Cosine groups its own kind together first. The sign flips.
   
   Function: Tangent
   Identity: \(\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}\)
   The “Memory Trick”: “The Fraction”: The numerator’s sign matches the original operation.
    
   

The Three-Step Process
To apply these identities, follow this workflow:
Step 1: Deconstruct the Angle
Look at your target angle and find two angles from the Unit Circle (30°, 45°, 60°, 90°, …) that add or subtract to equal it.
To get 15°: Use 45° - 30° or 60° - 45°.
To get 75°: Use 45° + 30°.
To get 105°: Use 60° + 45°.
Step 2: Expand the Identity
Plug your chosen angles into the correct formula.
Example: For sin(75°), use A=45° and B=30°.
sin(45 + 30) = sin(45°)cos(30°) + cos(45°)sin(30°)
Step 3: Substitute and Simplify
Replace the trig terms with their exact radical values from the Unit Circle.
sin(45°) = \(\frac{\sqrt{2}}{2}\)
cos(30°) = \(\frac{\sqrt{3}}{2}\)
cos(45°) = \(\frac{\sqrt{2}}{2}\)
sin(30°) = \(\frac{1}{2}\)
Calculation:
\(\left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}\)
 

Working in Reverse (Simplification)
Sometimes you will see a long, expanded expression and be asked to simplify it.
Problem: Simplify cos(80°)cos(20°) + sin(80°)sin(20°).
Identify the Pattern: It starts with \cos \cos. This is a Cosine identity.
Check the Sign: There is a plus in the middle. Since Cosine flips the sign, the original operation was subtraction.
Collapse: cos(80° - 20°) = cos(60°).
Solve: cos(60°) = \(\frac{1}{2}\).
 

Common Pitfalls to Avoid
Forgetting the Cosine Sign Flip: Many students write cos(A+B) = cos A cos B + sin A sin B$. Remember: Cosine is “contrary”—if you add the angles, you subtract the products.
Distributing the Function: Never do this: sin(45+30) = sin(45) + sin(30). This is a major mathematical error; the identities exist because functions don’t distribute over addition.
 

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

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