These formulas express trigonometric functions of multiples of an angle (like 2α or 3α) in terms of functions of the angle α itself. They are useful for simplifying expressions and solving equations involving multiple angles.
\[ \sin 2\alpha = 2 \sin \alpha \cos \alpha \]
Explanation: The double-angle formula for sine expresses \(\sin 2\alpha\) as twice the product of \(\sin \alpha\) and \(\cos \alpha\). It arises from the angle addition formula \(\sin(\alpha + \alpha)\).
\[ \cos 2\alpha = 2 \cos^2 \alpha - 1 = 1 - 2 \sin^2 \alpha = \cos^2 \alpha - \sin^2 \alpha \]
Explanation: The double-angle formula for cosine has three equivalent forms. They come from the cosine addition formula and the Pythagorean identity, allowing flexibility depending on known values.
\[ \tan 2\alpha = \frac{2 \tan \alpha}{1 - \tan^2 \alpha} \]
Explanation: The double-angle formula for tangent expresses \(\tan 2\alpha\) in terms of \(\tan \alpha\), derived using tangent addition formulas.
\[ \cot 2\alpha = \frac{\cot^2 \alpha - 1}{2 \cot \alpha} = \frac{\cot \alpha - \tan \alpha}{2} \]
Explanation: The double-angle formula for cotangent is less common but can be expressed in two equivalent ways. It relates cotangent of double angle to cotangent and tangent of \(\alpha\).
\[ \sin 3\alpha = 3 \sin \alpha - 4 \sin^3 \alpha \]
Explanation: The triple-angle formula for sine expresses \(\sin 3\alpha\) using powers of \(\sin \alpha\). It is derived from angle addition and power-reduction identities.
\[ \cos 3\alpha = 4 \cos^3 \alpha - 3 \cos \alpha \]
Explanation: The triple-angle formula for cosine expresses \(\cos 3\alpha\) as a cubic polynomial in \(\cos \alpha\).
\[ \tan 3\alpha = \frac{3 \tan \alpha - \tan^3 \alpha}{1 - 3 \tan^2 \alpha} \]
Explanation: The triple-angle formula for tangent expresses \(\tan 3\alpha\) in terms of \(\tan \alpha\), useful for solving complex angle equations.
\[ \cot 3\alpha = \frac{\cot^3 \alpha - \cot \alpha}{3 \cot^2 \alpha - 1} \]
Explanation: The triple-angle formula for cotangent relates \(\cot 3\alpha\) to powers of \(\cot \alpha\).