Math 140
Class 13
Summary of Special Values
| \(t\) | \(\sin t\) | \(\cos t\) | \(\tan t\) |
|---|---|---|---|
| \(0\) | \(0\) | \(1\) | \(0\) |
| \(\frac{\pi}{6}\) | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{3}}{3}\) |
| \(\frac{\pi}{4}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(1\) |
| \(\frac{\pi}{3}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\sqrt{3}\) |
| \(\frac{\pi}{2}\) | \(1\) | \(0\) | undefined |
| quadrant | \(\sin t\) | \(\cos t\) | \(\tan t\) |
|---|---|---|---|
| I | \(+\) | \(+\) | \(+\) |
| II | \(+\) | \(-\) | \(-\) |
| III | \(-\) | \(-\) | \(+\) |
| IV | \(-\) | \(+\) | \(-\) |
Tangent and Cotangent
\[\begin{align*} y &= ax\\ x &= \frac{1}{a}y\\ a &= \tan t\\ \frac{1}{a} &= \cot t \end{align*}\]
\(\color{red}{\tan t = \frac{y}{x} = \frac{\sin t}{\cos t}}\)
\(\color{red}{\cot t = \frac{x}{y} = \frac{\cos t}{\sin t}}\)
Secant and Cosecant
Summary
\(\displaystyle \tan t = \frac{\sin t}{\cos t}\)
\(\displaystyle \cot t = \frac{\cos t}{\sin t}\)
\(\displaystyle \sec t = \frac{1}{\cos t}\)
\(\displaystyle \csc t = \frac{1}{\sin t}\)
\[\cos^2 t + \sin^2 t = 1\]
Complex Unit Circle
\(\operatorname{cis}(s + t) = \operatorname{cis}(s)\operatorname{cis}(t)\)
\[ \operatorname{cis}(s + t) = \color{blue}{\cos(s+t)} + i\color{green}{\sin(s+t)} \]
\[\begin{align*} \operatorname{cis}(s)\operatorname{cis}(t) &= \left(\cos s + i\sin s\right)\left(\cos t + i\sin t\right)\\ &= \cos s \cos t + i\cos s\sin t + i\sin s\cos t + \color{red}{i^2} \sin s\sin t\\ &= \cos s \cos t + i\cos s\sin t + i\sin s\cos t \color{red}{{}-{}} \sin s\sin t\\ &= \color{blue}{\cos s \cos t \color{red}{{}-{}} \sin s\sin t} + i\left(\color{green}{\cos s\sin t + \sin s\cos t}\right) \end{align*}\]
\[\begin{align*} \color{blue}{\cos(s+t)} &= \color{blue}{\cos s \cos t \color{red}{{}-{}} \sin s\sin t}\\ \color{green}{\sin(s+t)} &= \color{green}{\cos s\sin t + \sin s\cos t} \end{align*}\]
Addition formulas
\[\begin{align*} \cos(s+t) &= \cos s \cos t - \sin s\sin t\\ \sin(s+t) &= \cos s\sin t + \sin s\cos t \end{align*}\]
Subtraction Formulas
Double Angle Formulas
Example
\(\displaystyle\cos\left(\frac{5\pi}{12}\right)\)
Example
If \(s\) is in quadrant IV with \(\cos s = \frac{5}{7}\), and \(\frac{\pi}{2} \le t \le \pi\) with \(\sin t = \frac{\sqrt{5}}{3}\), find \(\cos(s + t)\).
Example
If \(\tan t = \frac{24}{7}\) and \(t\) is in the third quadrant, find the values of the other 5 trigonometric functions.