The double angle formulas for sine and cosine are powerful tools in trigonometry, allowing us to express $$\sin(2\theta)$$ and $$\cos(2\theta)$$ in terms of $$\sin(\theta)$$ and $$\cos(\theta)$$. These formulas are not just mathematical curiosities; they have practical applications in physics, engineering, and computer science. In this article, we will derive these formulas from first principles.

## The Double Angle Formula for Sine

The double angle formula for sine is:

$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$

### Derivation

To derive this formula, we start with the sum formula for sine:

$\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$

Setting $$A = B = \theta$$, we get:

$\sin(2\theta) = \sin(\theta)\cos(\theta) + \cos(\theta)\sin(\theta)$

$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$

And thus, we have derived the double angle formula for sine.

## The Double Angle Formulas for Cosine

The double angle formulas for cosine are:

1. $$\cos(2\theta) = \cos^2(\theta) – \sin^2(\theta)$$
2. $$\cos(2\theta) = 2\cos^2(\theta) – 1$$
3. $$\cos(2\theta) = 1 – 2\sin^2(\theta)$$

### Derivation

$\cos(A + B) = \cos(A)\cos(B) – \sin(A)\sin(B)$

Setting $$A = B = \theta$$, we get:

$\cos(2\theta) = \cos^2(\theta) – \sin^2(\theta)$

This is our first formula. To derive the other two, we use the Pythagorean identity $$\sin^2(\theta) + \cos^2(\theta) = 1$$:

1. $$\cos^2(\theta) – \sin^2(\theta) = \cos^2(\theta) – (1 – \cos^2(\theta)) = 2\cos^2(\theta) – 1$$
2. $$\cos^2(\theta) – \sin^2(\theta) = (1 – \sin^2(\theta)) – \sin^2(\theta) = 1 – 2\sin^2(\theta)$$

## Conclusion

We have successfully derived the double angle formulas for sine and cosine. These formulas are not just theoretical constructs but have real-world applications in various fields. Understanding them deeply enriches our understanding of trigonometry and, by extension, the universe itself.