Double-Angle and Half-Angle Formulas

Here is a situation you have probably encountered: you know the sine and cosine of a certain angle, say 30 degrees, and now you need the sine of 60 degrees. You might think, “Well, 60 is just 30 doubled, so maybe I can just double the sine?” If only it were that simple. Unfortunately, $\sin(60°) \neq 2 \sin(30°)$. You can check: $\sin(30°) = 0.5$, so doubling would give you 1. But $\sin(60°) = \frac{\sqrt{3}}{2} \approx 0.866$. Not the same.

So if you cannot just double the trig value, what can you do? This is where double-angle formulas come in. They tell you exactly how to find the sine, cosine, or tangent of $2\theta$ when you know the trig values of $\theta$. And their cousins, the half-angle formulas, work in reverse: they let you find trig values for $\theta/2$ when you know those for $\theta$.

If these formulas seem intimidating, here is something reassuring: you already know everything you need. The double-angle formulas are just special cases of the sum formulas. Remember $\sin(A + B) = \sin A \cos B + \cos A \sin B$? What happens when $A = B$? You get the double-angle formula for sine. That is all there is to it. The half-angle formulas take a bit more work to derive, but once you have them, using them is straightforward.

Core Concepts

The Double-Angle Formula for Sine

Let us start with the sum formula for sine:

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

Now, what if $A$ and $B$ are the same angle? Let $A = B = \theta$:

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

Both terms on the right are identical, so we add them:

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

That is the double-angle formula for sine. Notice how elegant it is: to find the sine of double an angle, you multiply the sine and cosine of the original angle, then double the result.

The Double-Angle Formulas for Cosine

The same approach works for cosine. Start with:

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

Let $A = B = \theta$:

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

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

This is the first form of the double-angle formula for cosine. But here is where it gets interesting: because $\sin^2 \theta + \cos^2 \theta = 1$, we can rewrite this formula in two other ways.

Second form: Replace $\sin^2 \theta$ with $1 - \cos^2 \theta$:

$$\cos(2\theta) = \cos^2 \theta - (1 - \cos^2 \theta) = \cos^2 \theta - 1 + \cos^2 \theta = 2\cos^2 \theta - 1$$

Third form: Replace $\cos^2 \theta$ with $1 - \sin^2 \theta$:

$$\cos(2\theta) = (1 - \sin^2 \theta) - \sin^2 \theta = 1 - 2\sin^2 \theta$$

So we have three equivalent formulas for $\cos(2\theta)$:

$$\cos(2\theta) = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$$

Which one should you use? It depends on what you know. If you only have the cosine, use $2\cos^2 \theta - 1$. If you only have the sine, use $1 - 2\sin^2 \theta$. If you have both, any of them will work.

The Double-Angle Formula for Tangent

For tangent, we use the sum formula:

$$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$

With $A = B = \theta$:

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

This formula requires that $\tan \theta$ exists and that $\tan^2 \theta \neq 1$ (otherwise the denominator would be zero).

Power-Reducing Formulas

The double-angle formulas have a hidden superpower: they let you eliminate powers of sine and cosine. This turns out to be incredibly useful in calculus when you need to integrate expressions like $\sin^2 x$ or $\cos^4 x$.

Start with the double-angle formula $\cos(2\theta) = 1 - 2\sin^2 \theta$ and solve for $\sin^2 \theta$:

$$2\sin^2 \theta = 1 - \cos(2\theta)$$

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

Similarly, from $\cos(2\theta) = 2\cos^2 \theta - 1$:

$$2\cos^2 \theta = 1 + \cos(2\theta)$$

$$\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}$$

These are called power-reducing formulas because they reduce powers of trig functions to first-degree expressions. A squared trig function becomes a simple cosine of double the angle.

For tangent:

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

Half-Angle Formulas

Now we flip the script. Instead of finding trig values for $2\theta$ from $\theta$, we want to find values for $\theta/2$ from $\theta$. The key insight is that the power-reducing formulas are already half-angle formulas in disguise.

In the power-reducing formula $\sin^2 \theta = \frac{1 - \cos(2\theta)}{2}$, replace $\theta$ with $\frac{\alpha}{2}$:

$$\sin^2 \left(\frac{\alpha}{2}\right) = \frac{1 - \cos(\alpha)}{2}$$

Taking the square root:

$$\sin\left(\frac{\alpha}{2}\right) = \pm\sqrt{\frac{1 - \cos \alpha}{2}}$$

Similarly for cosine:

$$\cos\left(\frac{\alpha}{2}\right) = \pm\sqrt{\frac{1 + \cos \alpha}{2}}$$

And for tangent, there are actually three useful forms:

$$\tan\left(\frac{\alpha}{2}\right) = \pm\sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}} = \frac{1 - \cos \alpha}{\sin \alpha} = \frac{\sin \alpha}{1 + \cos \alpha}$$

The last two forms for tangent do not require choosing a sign, which makes them especially convenient.

Choosing the Correct Sign

Here is the critical part that trips people up: the half-angle formulas have a $\pm$ in front. How do you know which sign to use?

The answer depends on which quadrant the half-angle falls in:

• Quadrant I (0 to 90 degrees): All trig functions are positive
• Quadrant II (90 to 180 degrees): Only sine is positive
• Quadrant III (180 to 270 degrees): Only tangent is positive
• Quadrant IV (270 to 360 degrees): Only cosine is positive

So when you use a half-angle formula:

1. First, figure out where the original angle $\alpha$ is
2. Then determine where $\alpha/2$ must be
3. Finally, choose the sign based on whether that trig function is positive or negative in that quadrant

For example, if $\alpha = 300°$, then $\alpha/2 = 150°$, which is in Quadrant II. In Quadrant II, sine is positive and cosine is negative. So you would use the positive square root for $\sin(150°)$ and the negative square root for $\cos(150°)$.

Why These Formulas Matter

You might be wondering: why not just use a calculator? Here are a few reasons these formulas are worth knowing:

1. Exact values: A calculator gives you decimals. These formulas give you exact expressions like $\frac{\sqrt{6} - \sqrt{2}}{4}$, which are more precise and often required in mathematics.

2. Calculus: When you learn integration, you will need to integrate things like $\sin^2 x$. You cannot integrate that directly, but you can integrate $\frac{1 - \cos(2x)}{2}$. Power-reducing formulas make this possible.

3. Understanding: Seeing how these formulas connect to each other helps you understand trigonometry as a coherent system, not just a collection of random rules.

Notation and Terminology

Term	Meaning	Example
Double-angle	Formula for $2\theta$ in terms of $\theta$	$\sin 2\theta = 2 \sin \theta \cos \theta$
Half-angle	Formula for $\theta/2$ in terms of $\theta$	$\cos(\theta/2) = \pm\sqrt{\frac{1 + \cos \theta}{2}}$
Power-reducing	Formula to rewrite powers of trig functions	$\sin^2\theta = \frac{1 - \cos 2\theta}{2}$

Examples

Key Properties and Rules

Double-Angle Formulas

$$\sin 2\theta = 2\sin\theta\cos\theta$$

$$\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$$

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

Power-Reducing Formulas

$$\sin^2\theta = \frac{1 - \cos 2\theta}{2}$$

$$\cos^2\theta = \frac{1 + \cos 2\theta}{2}$$

$$\tan^2\theta = \frac{1 - \cos 2\theta}{1 + \cos 2\theta}$$

Half-Angle Formulas

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

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

$$\tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}$$

Sign Selection for Half-Angle Formulas

Quadrant of $\theta/2$	$\sin(\theta/2)$	$\cos(\theta/2)$	$\tan(\theta/2)$
I (0 to 90 degrees)	+	+	+
II (90 to 180 degrees)	+	-	-
III (180 to 270 degrees)	-	-	+
IV (270 to 360 degrees)	-	+	-

Common Mistakes to Avoid

• Doubling the trig value: $\sin 2\theta \neq 2\sin\theta$. The correct formula is $\sin 2\theta = 2\sin\theta\cos\theta$.
• Forgetting to determine the sign: The half-angle formulas have $\pm$, and the sign depends on which quadrant $\theta/2$ is in.
• Using the wrong form of cos 2θ: Choose the form that matches what you know. If you only have sine, use $1 - 2\sin^2\theta$.
• Squaring when you should not: In half-angle formulas, remember that $\sin(\theta/2)$, not $\sin^2(\theta/2)$, equals the right side.

Real-World Applications

Projectile Motion

When you throw a ball at an angle $\theta$, the range (horizontal distance) depends on $\sin 2\theta$. The range formula is:

$$R = \frac{v_0^2 \sin 2\theta}{g}$$

where $v_0$ is the initial velocity and $g$ is gravitational acceleration. This explains why 45 degrees gives maximum range: $\sin(2 \times 45°) = \sin 90° = 1$, the largest possible value.

Optics and Light Interference

When light waves interfere, the intensity patterns involve expressions like $\cos^2\theta$. Power-reducing formulas convert these to forms that are easier to analyze and integrate, helping physicists understand phenomena like thin-film interference (the rainbow colors on soap bubbles).

Signal Processing and Modulation

Radio and communication signals often involve products of sine and cosine waves. The identity $\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$ (related to double-angle formulas) is essential for understanding amplitude modulation (AM radio) and frequency mixing in electronics.

Simplifying Physics Calculations

Many physics problems become simpler with these formulas. For instance, the kinetic energy of a rotating object involves $\sin^2\theta$ and $\cos^2\theta$ terms. Converting these using power-reducing formulas often reveals underlying patterns or makes integration possible.

Self-Test Problems

Summary

• The double-angle formulas let you find trig values of $2\theta$ when you know trig values of $\theta$. The key formulas are:

  • $\sin 2\theta = 2\sin\theta\cos\theta$
  • $\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
  • $\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$

• The three forms of cos 2θ give you flexibility: use the form that matches what information you have.

• Power-reducing formulas rewrite squared trig functions in terms of double angles:

  • $\sin^2\theta = \frac{1 - \cos 2\theta}{2}$
  • $\cos^2\theta = \frac{1 + \cos 2\theta}{2}$

  These are essential for calculus integration.

• The half-angle formulas find trig values of $\theta/2$ from $\theta$:

  • $\sin(\theta/2) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$
  • $\cos(\theta/2) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$
  • $\tan(\theta/2) = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}$

• For half-angle formulas, choose the sign based on which quadrant $\theta/2$ falls in. The tangent forms without the $\pm$ can make life easier.

• These formulas are derived from sum formulas and the Pythagorean identity. You do not need to memorize them from scratch; you can derive them if you remember where they come from.

• Real-world applications include projectile motion (range depends on $\sin 2\theta$), optics, signal processing, and simplifying physics calculations.

• The fundamental skill is recognizing when to apply these formulas: when you see $2\theta$ or $\theta/2$, or when you need to reduce powers of trig functions, these are your tools.