Half-angle formulas express trigonometric functions of half an angle in terms of the cosine or sine of the full angle. These formulas simplify computations in integration, solving equations, and geometric problems.
\[ \sin \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{2}} \]
Explanation: Sine of half an angle relates to cosine of the full angle. The sign depends on the quadrant of \(\frac{\alpha}{2}\).
\[ \cos \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos \alpha}{2}} \]
Explanation: Cosine of half an angle is expressed using cosine of the full angle, with sign depending on the quadrant.
\[ \tan \frac{\alpha}{2} = \frac{\sin \alpha}{1 + \cos \alpha} = \frac{1 - \cos \alpha}{\sin \alpha} \]
Explanation: Tangent of half an angle can be written in two equivalent forms involving sine and cosine of the full angle.
\[ \cot \tan \frac{\alpha}{2} = \frac{\sin \alpha}{1 - \cos \alpha} = \frac{1 + \cos \alpha}{\sin \alpha} \]
Explanation: Cotangent of tangent of half an angle, a more advanced identity connecting these nested functions.
\[ \sin \alpha = \frac{2 \tan \frac{\alpha}{2}}{1 + \tan^2 \frac{\alpha}{2}} \]
Explanation: Expresses sine of full angle in terms of tangent of half angle.
\[ \cos \alpha = \frac{1 - \tan^2 \frac{\alpha}{2}}{1 + \tan^2 \frac{\alpha}{2}} \]
Explanation: Cosine of full angle in terms of tangent of half angle.
\[ \tan \alpha = \frac{2 \tan \frac{\alpha}{2}}{1 - \tan^2 \frac{\alpha}{2}} \]
Explanation: Tangent of full angle expressed through tangent of half angle.
\[ | \cos \alpha \pm \sin \alpha | = \sqrt{1 + \sin 2\alpha} \]
Explanation: An identity combining sine and cosine with double angle sine.
\[ 1 + \cos \alpha = 2 \cos^2 \frac{\alpha}{2} \]
Explanation: Relates cosine of angle to cosine squared of half angle.
\[ 1 - \cos \alpha = 2 \sin^2 \frac{\alpha}{2} \]
Explanation: Relates cosine of angle to sine squared of half angle.
\[ 1 + \sin \alpha = 2 \cos^2 \left(\frac{\pi}{4} - \frac{\alpha}{2}\right) \]
Explanation: Expression for \(1 + \sin \alpha\) using shifted cosine squared.
\[ 1 - \sin \alpha = 2 \sin^2 \left(\frac{\pi}{4} - \frac{\alpha}{2}\right) \]
Explanation: Expression for \(1 - \sin \alpha\) using shifted sine squared.