Sine, Cosine, and Tangent of a Double Angle
Big Idea: The double angle formulas allow us to express trigonometric functions of $2\theta$ in terms of trigonometric functions of $\theta$. These are powerful tools that simplify complex expressions and solve equations that would otherwise be very difficult.
Where Do Double Angle Formulas Come From?
Recall the angle addition formulas:
$$ \sin(A + B) = \sin A \cos B + \cos A \sin B $$
$$ \cos(A + B) = \cos A \cos B - \sin A \sin B $$
$$ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} $$
Now, what happens when we set $A = B = \theta$? We get $A + B = 2\theta$, and the formulas become the double angle formulas!