Double-angle and half-angle formulas are trigonometric identities that rewrite expressions involving
\(2x\) or \(\frac{x}{2}\). They are useful for exact values, simplification, and solving equations.
1. Double-angle formulas
\[
\sin(2x)=2\sin x\cos x
\]
\[
\cos(2x)=\cos^2x-\sin^2x
\]
The cosine formula can also be written in two very useful forms:
\[
\cos(2x)=1-2\sin^2x
\]
\[
\cos(2x)=2\cos^2x-1
\]
\[
\tan(2x)=\frac{2\tan x}{1-\tan^2x}
\]
2. Half-angle formulas
\[
\sin\left(\frac{x}{2}\right)=\pm\sqrt{\frac{1-\cos x}{2}}
\]
\[
\cos\left(\frac{x}{2}\right)=\pm\sqrt{\frac{1+\cos x}{2}}
\]
\[
\tan\left(\frac{x}{2}\right)=\pm\sqrt{\frac{1-\cos x}{1+\cos x}}
\]
The sign depends on the quadrant of \(\frac{x}{2}\), not the quadrant of \(x\).
3. Example: \(\cos(2x)\) from \(\sin x=\frac35\)
Use:
\[
\cos(2x)=1-2\sin^2x
\]
Substitute \(\sin x=\frac35\):
\[
\cos(2x)=1-2\left(\frac35\right)^2
\]
\[
\cos(2x)=1-2\cdot\frac9{25}
\]
\[
\cos(2x)=1-\frac{18}{25}
\]
\[
\cos(2x)=\frac7{25}
\]
4. Why signs matter
Some double-angle values are sign-dependent. For example:
\[
\sin(2x)=2\sin x\cos x
\]
If only \(\sin x\) is known, then \(\cos x\) may be positive or negative depending on the quadrant.
Therefore, \(\sin(2x)\) may have two possible signs.
5. Formula summary
| Formula type |
Identity |
Important note |
| Double sine |
\(\sin(2x)=2\sin x\cos x\) |
Needs signs of both \(\sin x\) and \(\cos x\) |
| Double cosine |
\(\cos(2x)=1-2\sin^2x=2\cos^2x-1\) |
Often works from only \(\sin x\) or only \(\cos x\) |
| Double tangent |
\(\tan(2x)=\dfrac{2\tan x}{1-\tan^2x}\) |
Undefined when \(1-\tan^2x=0\) |
| Half sine |
\(\sin\left(\frac{x}{2}\right)=\pm\sqrt{\frac{1-\cos x}{2}}\) |
Sign depends on the quadrant of \(\frac{x}{2}\) |
| Half cosine |
\(\cos\left(\frac{x}{2}\right)=\pm\sqrt{\frac{1+\cos x}{2}}\) |
Sign depends on the quadrant of \(\frac{x}{2}\) |
6. Common mistakes
- Do not write \(\sin(2x)=2\sin x\). The correct formula is \(2\sin x\cos x\).
- Do not write \(\cos(2x)=2\cos x\). Use one of the cosine double-angle identities.
- For half-angle formulas, always choose the sign from the quadrant of \(\frac{x}{2}\).
- For \(\tan(2x)\), check the denominator \(1-\tan^2x\).
- Keep angle units consistent: degrees with degrees, radians with radians.
