Trigonometry Equation Sheet
Basic Trig Identities
\[
\sin(x)=\frac{1}{\csc(x)}
\]
\[
\cos(x)=\frac{1}{\sec(x)}
\]
\[
\tan(x)=\frac{\sin(x)}{\cos(x)}
\]
\[
\cot(x)=\frac{1}{\tan(x)}
\]
\[
\sin(-x)=-\sin(x)
\]
\[
\cos(-x)=\cos(x)
\]
\[
\tan(-x)=-\tan(x)
\]
\[
\sin^2(x)+\cos^2(x)=1
\]
\[
\tan^2(x)+1=\sec^2(x)
\]
\[
\cot^2(x)+1=\csc^2(x)
\]
Half Angle
\[
\sin\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1-\cos(\theta)}{2}}
\]
\[
\cos\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1+\cos(\theta)}{2}}
\]
\[
\tan\left(\frac{\theta}{2}\right)=\frac{\sin(\theta)}{1+\cos(\theta)}=\frac{1-\cos(\theta)}{\sin(\theta)}
\]
Power Reduction
\[
\sin^2(\theta)=\frac{1}{2}(1-\cos(2\theta))
\]
\[
\cos^2(\theta)=\frac{1}{2}(1+\cos(2\theta))
\]
\[
\tan^2(\theta)=\frac{1-\cos(2\theta)}{1+\cos(2\theta)}
\]
Angle Sum and Difference
\[
\sin(\theta\pm\phi)=\sin(\theta)\cos(\phi)\pm\cos(\theta)\sin(\phi)
\]
\[
\cos(\theta\pm\phi)=\cos(\theta)\cos(\phi)\mp\sin(\theta)\sin(\phi)
\]
\[
\tan(\theta\pm\phi)=\frac{\tan(\theta)\pm\tan(\phi)}{1\mp\tan(\theta)\tan(\phi)}
\]
Double Angle
\[
\sin(2\theta)=2\sin(\theta)\cos(\theta)
\]
\[
\cos(2\theta)=\cos^2(\theta)-\sin^2(\theta)
\]
\[
\tan(2\theta)=\frac{2\tan(\theta)}{1-\tan^2(\theta)}
\]
Sum to Product
\[
\sin(\theta)+\sin(\phi)=2\sin\left(\frac{\theta+\phi}{2}\right)\cos\left(\frac{\theta-\phi}{2}\right)
\]
\[
\sin(\theta)-\sin(\phi)=2\cos\left(\frac{\theta+\phi}{2}\right)\sin\left(\frac{\theta-\phi}{2}\right)
\]
\[
\cos(\theta)+\cos(\phi)=2\cos\left(\frac{\theta+\phi}{2}\right)\cos\left(\frac{\theta-\phi}{2}\right)
\]
\[
\cos(\theta)-\cos(\phi)=-2\sin\left(\frac{\theta+\phi}{2}\right)\sin\left(\frac{\theta-\phi}{2}\right)
\]