Double-Angle Identities Formula
The Formula
\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta
\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}
When to use: What if both angles in the sum formula are the same? Setting A = B = \theta in the sum identities gives you the double-angle formulas. They answer: if you know the trig values for an angle, what are the trig values for twice that angle? The cosine double-angle formula is especially versatile because it has three equivalent forms, each useful in different situations—pick whichever one simplifies your problem.
Quick Example
Confirm: \sin 60° = \frac{\sqrt{3}}{2}. \checkmark
Notation
What This Formula Means
Formulas expressing \sin(2\theta), \cos(2\theta), and \tan(2\theta) in terms of single-angle trig functions.
What if both angles in the sum formula are the same? Setting A = B = \theta in the sum identities gives you the double-angle formulas. They answer: if you know the trig values for an angle, what are the trig values for twice that angle? The cosine double-angle formula is especially versatile because it has three equivalent forms, each useful in different situations—pick whichever one simplifies your problem.
Formal View
Worked Examples
Example 1
easySolution
- 1 Use the double-angle formula: \cos(2\theta) = 2\cos^2(\theta) - 1.
- 2 Substitute: \cos(2\theta) = 2\left(\frac{3}{5}\right)^2 - 1 = 2 \cdot \frac{9}{25} - 1.
- 3 = \frac{18}{25} - 1 = \frac{18}{25} - \frac{25}{25} = -\frac{7}{25}.
Answer
Example 2
mediumCommon Mistakes
- Writing \sin(2\theta) = 2\sin\theta instead of 2\sin\theta\cos\theta—you can't just double the sine; the cosine factor is essential.
- Using the wrong form of \cos(2\theta) and making the problem harder—choose the form that matches the information you have.
- Forgetting the restriction on \tan(2\theta): the formula is undefined when \tan^2\theta = 1, i.e., when \theta = \frac{\pi}{4} + \frac{n\pi}{2}.
Why This Formula Matters
These formulas are essential in calculus (integration of \sin^2 x and \cos^2 x), physics (wave interference, power in AC circuits), and simplifying trigonometric expressions. The rearranged forms also yield the power-reduction and half-angle formulas.
Frequently Asked Questions
What is the Double-Angle Identities formula?
Formulas expressing \sin(2\theta), \cos(2\theta), and \tan(2\theta) in terms of single-angle trig functions.
How do you use the Double-Angle Identities formula?
What if both angles in the sum formula are the same? Setting A = B = \theta in the sum identities gives you the double-angle formulas. They answer: if you know the trig values for an angle, what are the trig values for twice that angle? The cosine double-angle formula is especially versatile because it has three equivalent forms, each useful in different situations—pick whichever one simplifies your problem.
What do the symbols mean in the Double-Angle Identities formula?
The three forms of \cos(2\theta) are all equivalent. Use \cos^2\theta - \sin^2\theta when you have both; 2\cos^2\theta - 1 when you only know cosine; 1 - 2\sin^2\theta when you only know sine.
Why is the Double-Angle Identities formula important in Math?
These formulas are essential in calculus (integration of \sin^2 x and \cos^2 x), physics (wave interference, power in AC circuits), and simplifying trigonometric expressions. The rearranged forms also yield the power-reduction and half-angle formulas.
What do students get wrong about Double-Angle Identities?
Students often forget there are three forms of \cos(2\theta). The choice of form depends on context—pick the one that eliminates the trig function you don't need.
What should I learn before the Double-Angle Identities formula?
Before studying the Double-Angle Identities formula, you should understand: trig identities sum difference.