Double Angle Formula Calculator

What are Double Angle Formulas?

In trigonometry, double angle formulas (or double angle identities) relate the trigonometric functions of double angles ($2\theta$) to functions of the single angle ($\theta$). These identities are special cases of the angle addition formulas $\sin(\alpha + \beta)$, $\cos(\alpha + \beta)$, and $\tan(\alpha + \beta)$ where $\alpha = \beta = \theta$.

Primary Double Angle Identities

1. Sine Double Angle Formula

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

Derivation: Using the sum identity $\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)$ and setting $\alpha = \beta = \theta$, we obtain $\sin(2\theta) = \sin(\theta)\cos(\theta) + \cos(\theta)\sin(\theta) = 2\sin(\theta)\cos(\theta)$.

2. Cosine Double Angle Formulas

The cosine double angle formula has three equivalent algebraic forms derived using the fundamental Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$:

• Standard Form: $$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$$
• In terms of Cosine only: $$\cos(2\theta) = 2\cos^2(\theta) - 1$$
• In terms of Sine only: $$\cos(2\theta) = 1 - 2\sin^2(\theta)$$

3. Tangent Double Angle Formula

$$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$$

Provided that $\tan(\theta) \neq \pm 1$ and $\theta \neq \frac{\pi}{4} + \frac{k\pi}{2}$.

4. Reciprocal Double Angle Identities

• $$\csc(2\theta) = \frac{1}{\sin(2\theta)} = \frac{1}{2\sin(\theta)\cos(\theta)}$$
• $$\sec(2\theta) = \frac{1}{\cos(2\theta)} = \frac{1}{\cos^2(\theta) - \sin^2(\theta)}$$
• $$\cot(2\theta) = \frac{1}{\tan(2\theta)} = \frac{\cot^2(\theta) - 1}{2\cot(\theta)}$$

Step-by-Step Example: Given a Trigonometric Ratio

Problem: Given that $\sin(\theta) = \frac{3}{5}$ and $\theta$ is in Quadrant II, find $\sin(2\theta)$, $\cos(2\theta)$, and $\tan(2\theta)$.

1. Find $\cos(\theta)$ using Pythagorean Identity:
   Since $\theta$ is in Quadrant II, $\cos(\theta)$ is negative.
   $$\cos(\theta) = -\sqrt{1 - \sin^2(\theta)} = -\sqrt{1 - \left(\frac{3}{5}\right)^2} = -\sqrt{\frac{16}{25}} = -\frac{4}{5}$$
2. Calculate $\sin(2\theta)$::
   $$\sin(2\theta) = 2 \sin(\theta) \cos(\theta) = 2 \left(\frac{3}{5}\right) \left(-\frac{4}{5}\right) = -\frac{24}{25} = -0.96$$
3. Calculate $\cos(2\theta)$::
   $$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = \left(-\frac{4}{5}\right)^2 - \left(\frac{3}{5}\right)^2 = \frac{16}{25} - \frac{9}{25} = \frac{7}{25} = 0.28$$
4. Calculate $\tan(2\theta)$::
   $$\tan(2\theta) = \frac{\sin(2\theta)}{\cos(2\theta)} = \frac{-24/25}{7/25} = -\frac{24}{7} \approx -3.42857$$

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