13.2: Trigonometric Identities

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Sine and cosine are periodic, leading to the following identities:

\[\sin\theta =-\sin\left ( -\theta \right )=-\cos(\theta +\frac{\pi }{2})=\cos(\theta -\frac{\pi }{2})\]

\[\cos\theta =\cos\left ( -\theta \right )=\sin(\theta +\frac{\pi }{2})=-\sin(\theta -\frac{\pi }{2})\]

The sine or cosine for sums or differences between angles can be calculated using the following identities:

\[\cos(\theta _{1}+\theta _{2})=\cos(\theta _{1})\cos(\theta _{2})-\sin(\theta _{1})\sin(\theta _{2})\]

\[\sin(\theta _{1}+\theta _{2})=\sin(\theta _{1})\cos(\theta _{2})+\cos(\theta _{1})\sin(\theta _{2})\]

\[\cos(\theta _{1}-\theta _{2})=\cos(\theta _{1})\cos(\theta _{2})+\sin(\theta _{1})\sin(\theta _{2})\]

\[\sin(\theta _{1}-\theta _{2})=\sin(\theta _{1})\cos(\theta _{2})-\cos(\theta _{1})\sin(\theta _{2})\]

The sum of the squares of sine and cosine for the same angle is one:

\[\cos(\theta )\cos(\theta )+\sin(\theta )\sin(\theta )=1\]