Double and Half Angle Formulas
The double and half angle formulas let you find exact values of trig functions at angles you might not have memorized. If you know the common values — sin(30°), cos(45°), and so on — you can use these formulas to work out angles like 15°, 60°, or 120° without a calculator. They also show up frequently in calculus, where they're used to simplify integrals that would otherwise be very difficult to solve.
Double Angle Formulas
$$\sin(2\theta) = 2\sin\theta\cos\theta$$
$$\cos(2\theta) = \cos^2\theta - \sin^2\theta$$
There are two alternate forms of the cosine formula, derived by substituting from the Pythagorean identity \(\sin^2\theta = 1 - \cos^2\theta\) or \(\cos^2\theta = 1 - \sin^2\theta\):
$$\cos(2\theta) = 2\cos^2\theta - 1$$
$$\cos(2\theta) = 1 - 2\sin^2\theta$$
Having three versions is useful — you can pick whichever one fits what you already know.
$$\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$$
Example: Finding sin(120°)
We know sin(60°) = \(\frac{\sqrt{3}}{2}\) and cos(60°) = \(\frac{1}{2}\). Since 120° = 2 × 60°, we can use the double angle formula:
$$\sin(120°) = 2\sin(60°)\cos(60°) = 2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \frac{\sqrt{3}}{2}$$
That matches the known value of sin(120°), which confirms the formula works.
Example: Finding cos(60°) from sin(30°) alone
This shows why having multiple forms is handy. Using \(\cos(2\theta) = 1 - 2\sin^2\theta\) with \(\theta = 30°\):
$$\cos(60°) = 1 - 2\sin^2(30°) = 1 - 2\left(\frac{1}{2}\right)^2 = 1 - \frac{1}{2} = \frac{1}{2}$$
Half Angle Formulas
$$\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}$$
$$\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}$$
$$\tan\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} = \frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}$$
The ± Sign
The \(\pm\) doesn't mean you use both signs — it means you have to figure out which one applies. The sign depends on which quadrant \(\frac{\theta}{2}\) lands in, not \(\theta\) itself.
For example, if \(\theta = 200°\), then \(\frac{\theta}{2} = 100°\), which is in the second quadrant where sine is negative. But if \(\theta = 30°\), then \(\frac{\theta}{2} = 15°\), which is in the first quadrant where sine is positive. Sketch a quick unit circle if you're not sure which quadrant you're in.
Example: Finding sin(15°)
Since 15° = 30°/2, use the half angle formula with \(\theta = 30°\) and cos(30°) = \(\frac{\sqrt{3}}{2}\):
$$\sin(15°) = \sqrt{\frac{1 - \cos(30°)}{2}} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{\sqrt{2 - \sqrt{3}}}{2}$$
The positive root is correct because 15° is in the first quadrant. Numerically this works out to about 0.259, which you can verify with a calculator.
Putting It Together
The double and half angle formulas are really just specialized versions of the addition formulas — sin(2θ) is just sin(θ + θ) expanded and simplified. So there's nothing magic about them; they're consequences of identities you may have already seen.
Where they're most useful is in situations where you need an exact value at an angle that isn't on the standard unit circle — 15°, 22.5°, 75°, and so on. Start from the nearest familiar angle, decide whether to double or halve, and apply the right formula. For the half angle formulas, always establish the sign first before plugging in numbers.