Module 4: Basic Facts

2.1 Angles

It is useful to know the values of trigonometric functions for some well-known angles.

$\begin{array}{ccccc} \sin (θ) & \cos (θ) & \tan (θ) & \cot (θ) \\ 0 & 0 & 1 & 0 & Undefined \\ \frac{π}{6} (30 degrees) & \frac{1}{2} & \frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{3} & \sqrt{3} \\ \frac{π}{4} (45 degrees) & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 1 & 1 \\ \frac{π}{3} (60 degrees) & \frac{\sqrt{3}}{2} & \frac{1}{2} & \sqrt{3} & \frac{\sqrt{3}}{3} \\ \frac{π}{2} (90 degrees) & 1 & 0 & Undefined & 0 \\ π (180 degrees) & 0 & - 1 & 0 & Undefined \\ \frac{3 π}{2} (270 degrees) & - 1 & 0 & Undefined & 0 \end{array}$ $\begin{array}{|c|c|c|c|c|}\hline &&&&\\ & \sin( \theta)& \cos( \theta) & \tan( \theta) & \cot ( \theta) \\ &&&&\\\hline &&&&\\ 0&0&1&0&\mbox{Undefined}\\ &&&&\\\hline &&&&\\ \frac{\pi}{6}\; (30\; \mbox{degrees)}& \frac{1}{2}& \frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{3}& \sqrt{3}\\ &&&&\\\hline &&&&\\ \frac{\pi}{4}\; (45\; \mbox{degrees)}& \frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}&1& 1\\&&&&\\ \hline &&&&\\ \frac{\pi}{3}\; (60\; \mbox{degrees)}& \frac{\sqrt{3}}{2}& \frac{1}{2}&\sqrt{3}& \frac{\sqrt{3}}{3}\\&&&&\\ \hline &&&&\\ \frac{\pi}{2}\; (90\; \mbox{degrees)}&1&0&\mbox{Undefined}&0\\&&&&\\\hline &&&&\\ \pi\; (180\; \mbox{degrees)}&0&-1&0&\mbox{Undefined}\\&&&&\\\hline &&&&\\ \frac{3\pi}{2}\; (270\; \mbox{degrees)}&-1&0&\mbox{Undefined}&0\\&&&&\\\hline \end{array}$

With the identities listed in the next section you will be able to compute the value of trigonometric functions for a few other angles.

2.2 Identities

Here are a list of some basic trigonometric identities:

$LaTeX: \sin^2 (x) + \cos^2 (x) =1$ $\sin^2 (x) + \cos^2 (x) =1$

$LaTeX: \cos(-x)=\cos(x),$ $\cos(-x)=\cos(x),$ Cosine is an even function.

$LaTeX: \sin(-x)=-\sin(x),$ $\sin(-x)=-\sin(x),$ Sine is an odd function.

$LaTeX: \cos(x+y)=\cos x \cos y - \sin x \sin y$ $\cos(x+y)=\cos x \cos y - \sin x \sin y$

$LaTeX: \sin(x+y)=\sin x \cos y + \cos x \sin y$ $\sin(x+y)=\sin x \cos y + \cos x \sin y$

2.3 Practice

Find a formula or value for $LaTeX: \sin(\frac{3\pi}{4}), \sin(x-y),$ $\sin(\frac{3\pi}{4}), \sin(x-y),$ and $LaTeX: \cos(2x).$ $\cos(2x).$

$LaTeX: \begin{align*} \sin\left(\frac{3\pi}{4}\right) &= \sin\left(\frac{\pi}{2}+\frac{\pi}{4}\right)\\ &= \sin\left(\frac{\pi}{2}\right) \cos\left(\frac{\pi}{4}\right) + \cos\left(\frac{\pi}{2}\right) \sin \left(\frac{\pi}{4}\right)\\ &= 1 \cdot \frac{\sqrt{2}}{2}+0 \cdot \frac{\sqrt{2}}{2}\\ &=\frac{\sqrt{2}}{2}\\ \\ \sin(x-y)&= \sin (x) \cos (y) - \cos (x) \sin( y)\\ \\ \cos(2x) &= \cos(x+x)\\ &= \cos (x) \cos x -\sin (x) \sin (x)\\ &=\cos^2 (x) -\sin^2 (x) \end{align*}$ $\begin{align*} \sin\left(\frac{3\pi}{4}\right) &= \sin\left(\frac{\pi}{2}+\frac{\pi}{4}\right)\\ &= \sin\left(\frac{\pi}{2}\right) \cos\left(\frac{\pi}{4}\right) + \cos\left(\frac{\pi}{2}\right) \sin \left(\frac{\pi}{4}\right)\\ &= 1 \cdot \frac{\sqrt{2}}{2}+0 \cdot \frac{\sqrt{2}}{2}\\ &=\frac{\sqrt{2}}{2}\\ \\ \sin(x-y)&= \sin (x) \cos (y) - \cos (x) \sin( y)\\ \\ \cos(2x) &= \cos(x+x)\\ &= \cos (x) \cos x -\sin (x) \sin (x)\\ &=\cos^2 (x) -\sin^2 (x) \end{align*}$

Remark: Note using the identity $LaTeX: \cos^2 (x)+ \sin ^2 (x) =1$ $\cos^2 (x)+ \sin ^2 (x) =1$ , we can rewrite $LaTeX: \cos(2x)$ $\cos(2x)$ .

$LaTeX: \begin{align*} \cos(2x) &=\cos^2 (x) -\sin^2(x)\\ &= 2\cos^2 (x) -1\\ &= 1-2 \sin^2 (x) \end{align*}$ $\begin{align*} \cos(2x) &=\cos^2 (x) -\sin^2(x)\\ &= 2\cos^2 (x) -1\\ &= 1-2 \sin^2 (x) \end{align*}$