Angle Identities Game

Angle Identities Quiz/Game
This Angle Identities game will test your knowledge of how trigonometric functions relate to each other algebraically. Scroll down the page for a more detailed explanation.
 

 

How to Play

1. Analyze the Prompt:
   Read the center text carefully. Note if it’s a “2” (Double) or a fraction (Half).
   
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).
    
   

Here are the Angle Identities use in this game.
Double-Angle Identities
These formulas describe what happens when you “double” the input angle (2θ). They are derived from the Sum and Difference formulas.
Sine Double-Angle
There is only one main form for sine:
\(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\)
Concept: To get the sine of a double angle, you multiply the sine and cosine of the single angle by 2.
Cosine Double-Angle
Cosine has three equivalent forms. You can choose the one that fits your known variables:
The Base Form: \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\)
Using Sine only: \(\cos(2\theta) = 1 - 2\sin^2(\theta)\)
Using Cosine only: \(\cos(2\theta) = 2\cos^2(\theta) - 1\)
Tangent Double-Angle: \(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}\)

 

Half-Angle Identities
These formulas allow you to find the trig values for \(\frac{\theta}{2}\). These are particularly useful for finding exact values of angles like \(15^\circ\) (which is half of \(30^\circ\)).
Sine and Cosine Half-Angle
Both of these involve a square root. Note the sign difference between them:
Sine: \(\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}\)
Cosine: \(\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}\)
The \(\pm\) Mystery: The formula itself doesn’t tell you if the answer is positive or negative. You must look at which quadrant the half-angle \(\frac{\theta}{2}\) falls into. For example, if \(\theta = 300^\circ\), then \(\frac{\theta}{2} = 150^\circ\). Since \(150^\circ\) is in Quadrant II, the sine would be positive and the cosine would be negative.
Tangent Half-Angle
Tangent has two versions that don’t require square roots, making them much easier to use:
\(\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)} \quad \text{or} \quad \frac{\sin(\theta)}{1 + \cos(\theta)}\)
 

These videos give a clear, step-by-step approach to learn trigonometric identities.

• Show Half-Angle Identities Video

• Show Double-Angle Identities Video

 

Try out our new and fun Fraction Concoction Game.

Add and subtract fractions to make exciting fraction concoctions following a recipe. There are four levels of difficulty: Easy, medium, hard and insane. Practice the basics of fraction addition and subtraction or challenge yourself with the insane level.

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