Addition formulas, double angle formulas, and other trigonometric identities

Trigonometry: Trigonometric functions

Addition formulas, double angle formulas, and other trigonometric identities

Addition formulas The following three addition formulas also belong to the basic trigonometric formulas, but you do not need to know them by heart:
\[\begin{aligned}
\sin (\alpha + \beta ) &= \sin(\alpha) \cos(\beta) + \cos(\alpha)\sin(\beta)\\ \\
\cos (\alpha + \beta ) &= \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) \\ \\
\tan (\alpha + \beta ) &= \frac{\tan(\alpha) +\tan(\beta)}{1 - \tan(\alpha) \tan(\beta)} \\ \\
\end{aligned}\]

Double angle formulas Replacing \(\beta\) by \(\alpha\) in addition formulas gives the following double angle formulas
\[\begin{aligned}
\sin(2 \alpha) &= 2 \sin(\alpha) \cos(\alpha)\\ \\
\cos(2 \alpha) &= \cos(\alpha)^2 - \sin(\alpha)^2\\ \\ &= 2\cos(\alpha)^2 - 1\\ \\ &= 1 - 2\sin(\alpha)^2\\ \\
\tan (2\alpha ) &= \frac{2\tan(\alpha)}{1 - \tan(\alpha)^2}
\end{aligned}\]
Only the first two double angle formulas you need to know by heart.

Subtraction formulas Replacing \(\beta\) by \(-\beta\) in addition formulas gives
\[\begin{aligned}
\sin (\alpha - \beta ) &= \sin(\alpha) \cos(\beta) - \cos(\alpha) \sin(\beta)\\ \\
\cos (\alpha - \beta ) &= \cos(\alpha) \cos (\beta) + \sin(\alpha) \sin(\beta)\\ \\
\tan (\alpha + \beta ) &= \frac{\tan(\alpha) -\tan(\beta)}{1+\tan(\alpha) \tan(\beta)}
\end{aligned}\]
You do not need to know the identities by heart, and also not the next product formulas, and sum and difference formulas.

Product formulas Combination of earlier formulas leads to the following product formulas for sine and cosine:
\[\begin{aligned}
\sin (\alpha) \cos(\beta ) &= \tfrac{1}{2}\bigl(\sin(\alpha+\beta) + \sin(\alpha-\beta)\bigr)\\ \\
\cos (\alpha) \cos(\beta ) &= \tfrac{1}{2}\bigl(\cos(\alpha+\beta) + \cos(\alpha-\beta)\bigr)\\ \\
\sin (\alpha) \sin(\beta ) &= \tfrac{1}{2}\bigl(\cos(\alpha+\beta) - \cos(\alpha-\beta)\bigr) \\ \\
\end{aligned}\]

Sum and difference formulas The following formulas for sum and difference of trigonometric functions can be derived:
\[\begin{aligned}
\sin (\alpha) + \sin(\beta ) &= 2\sin\biggl(\frac{\alpha+\beta}{2}\biggr) \cos\biggl(\frac{\alpha-\beta}{2}\biggr)\\ \\
\sin (\alpha) - \sin(\beta ) &= 2\sin\biggl(\frac{\alpha-\beta}{2}\biggr) \cos\biggl(\frac{\alpha+\beta}{2}\biggr)\\ \\
\cos (\alpha) + \cos(\beta ) &= 2\cos\biggl(\frac{\alpha+\beta}{2}\biggr) \cos\biggl(\frac{\alpha-\beta}{2}\biggr)\\ \\
\cos (\alpha) - \cos(\beta ) &= -2\sin\biggl(\frac{\alpha+\beta}{2}\biggr) \sin\biggl(\frac{\alpha-\beta}{2}\biggr)
\end{aligned}\]

Although Thomas Simpson (1710-1761) is the actual discoverer of the above formulas, they are also referred to as Mollweide's formulas. Karl Brandan Mollweide (1774-1825) was a German mathematician and physicist, and he published the formulas named after him in 1808, albeit in a slightly different form.

These formulas can also be used to compute special function values.

Use the addition formulas to calculate#\sin\!\left(\dfrac{\pi}{12}\right)#.

#\sin\left(\dfrac{\pi}{12}\right)=\dfrac{\sqrt{3}-1}{2\sqrt{2}}#

\begin{eqnarray*}
\sin\left(\frac{\pi}{12}\right)&=&\sin\left(\frac{\pi}{3}-\dfrac{\pi}{4}\right)\\
&=&\sin\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{4}\right)-\cos\left(\frac{\pi}{3}\right)\sin\left(\frac{\pi}{4}\right)\\
&=&\dfrac{\sqrt{3}}{2}\cdot\frac{1}{\sqrt{2}}-\dfrac{1}{2}\cdot\frac{1}{\sqrt{2}}\\
&=&\frac{\sqrt{3}-1}{2\sqrt{2}}\tiny.
\end{eqnarray*}Alternative ways to split one twelfth:\[\frac{1}{12}=\frac{1}{3}-\frac{1}{4}=\frac{1}{4}-\frac{1}{6}=\frac{5}{12}-\frac{1}{3}=\frac{7}{12}-\frac{1}{2}\tiny .\] You can solve this exercise in different ways, but the result is always the same.

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The Double Angle Formula (26:33)

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