When you start learning about trigonometric functions, it's important to know the differences between double angle and half angle formulas. These formulas help you make math problems simpler and solve equations that use trigonometric functions.

Double Angle Formulas

Double angle formulas are used to find trigonometric functions for double the angle. They are helpful when you want to calculate things like $\sin(2\theta)$ or $\cos(2\theta)$. Here are the most common double angle formulas:

1. Sine:
   $\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$

2. Cosine:
   $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$
   You can also write it as:
   $\cos(2\theta) = 2\cos^2(\theta) - 1$
   or
   $\cos(2\theta) = 1 - 2\sin^2(\theta)$

3. Tangent:
   $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$

Half Angle Formulas

Half angle formulas help you find trigonometric functions for half the angle. These formulas are great when you need to figure out values for angles like $\frac{\theta}{2}$. Here are the key half angle formulas:

1. Sine:
   $\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$

2. Cosine:
   $\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$

3. Tangent:
   $\tan\left(\frac{\theta}{2}\right) = \frac{\sin(\theta)}{1 + \cos(\theta)} \text{ or } \frac{1 - \cos(\theta)}{\sin(\theta)}$

Summary of Differences

1. Purpose:

   • Double angle formulas deal with $2\theta$. They help you simplify or find trigonometric functions at double the angle.
   • Half angle formulas focus on $\frac{\theta}{2}$. They make it easier to simplify at half the angle.

2. Applications:

   • Use double angle formulas for problems that require $2\theta$ evaluations.
   • Use half angle formulas for problems that need values at $\frac{\theta}{2}$.

Knowing these differences is important for understanding trigonometric functions in your studies!