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Review of Trigonometric, Logarithmic, and Exponential Functions

In this tutorial, we review trigonometric, logarithmic, and exponential functions with a focus on those properties which will be useful in future math and science applications.

Trigonometric Functions

Geometrically, there are two ways to describe trigonometric functions:

Polar Angle

$\begin{array}{l} x=\cos\theta\\ y=\sin\theta\\ ~\\ {\small\textrm{Measure }} \theta {\small\textrm{ in radians:}}\\ \theta =\frac{{\small\textrm{arc length}}}{{\small\textrm{radius}}}\\ ~\\ {\small\textrm{For example,}}\quad 180^{\circ}=\displaystyle\frac{\pi r}{r}=\pi {\small\textrm{ radians}}\\ ~\\ {\small\textrm{Radians}}=\displaystyle\frac{{\small\textrm{degrees}}}{180}\cdot \pi \end{array}$




Right Triangle

$\begin{array}{l} \sin\theta &=& \displaystyle \frac{{\small\textrm{opposite}}}{{\small\textrm{hypotenuse}}} &=& \frac{y}{r}\\~\\ \cos\theta &=& \displaystyle \frac{{\small\textrm{adjacent}}}{{\small\textrm{hypotenuse}}} &=& \frac{x}{r}\\~\\ \tan\theta &=& \displaystyle \frac{{\small\textrm{opposite}}}{{\small\textrm{adjacent}}} &=& \frac{y}{x}\\~\\ \csc\theta &=& \displaystyle \frac{1}{\sin\theta} &=& \frac{r}{y}\\~\\ \sec\theta &=& \displaystyle \frac{1}{\cos\theta} &=& \frac{r}{x}\\~\\ \cot\theta &=& \displaystyle \frac{1}{\tan\theta} &=& \frac{x}{y} \end{array}$

Evaluating Trigonometric Functions

$0 {\small\textrm{ rad}} $ $\pi/6 {\small\textrm{ rad}} $ $\pi/4 {\small\textrm{ rad}} $ $\pi/3 {\small\textrm{ rad}} $ $\pi/2 {\small\textrm{ rad}}$
$0^{\circ} $ $30^{\circ} $ $45^{\circ} $ $60^{\circ} $ $90^{\circ}$
$\sin\theta $ $0 $ $1/2 $ $\sqrt{2}/2 $ $\sqrt{3}/2 $ $1$
$\cos\theta $ $1 $ $\sqrt{3}/2 $ $\sqrt{2}/2 $ $1/2 $ $0$
$\tan\theta $ $0 $ $\sqrt{3}/3 $ $1 $ $\sqrt{3} $ ${\small\textrm{undefined}}$


$\begin{array}{lcr} \sin(-\theta) & = & -\sin\theta \\ \cos(-\theta)& = & \cos\theta \\ ~\\ \cos(\theta+\pi) & = & -\cos\theta \\ \sin(\theta+\pi) & = & -\sin\theta \end{array}$ $\begin{array}{lcr} \sin(\theta +\pi/2) & = & \cos\theta\\ \cos(\theta +\pi/2) & = & -\sin\theta\\ ~\\ \cos(\theta +2\pi) & = & \cos\theta\\ \sin(\theta +2\pi) & = & \sin\theta \end{array}$


Trigonometric Identities

We list here some of the most commonly used identities:

$\begin{array}{lcr} \textbf{1. }\cos^2\theta+\sin^2\theta=1 \\ \textbf{2. }\cos^2\theta =\displaystyle\frac{1}{2}[1+\cos(2\theta)] \\ \textbf{3. } \sin^2\theta=\displaystyle\frac{1}{2}[1-\cos(2\theta)]\\ \textbf{4. } \sin(2\theta)=2\sin\theta\cos\theta \\ \textbf{5. }\cos(2\theta)=\cos^2\theta-\sin^2\theta \end{array}$ $\begin{array}{lcr} \textbf{6. } \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\\ \textbf{7. } \cos(\alpha +\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\\ ~\\ \textbf{8. } C_1\cos(\omega x)+C_2\sin(\omega x)=A\sin(\omega x+\phi)\\ \qquad{\small\textrm{where }} A=\sqrt{C_1^2+C_2^2},\quad \phi=\arctan (C_1/C_2) \end{array}$

Graphs of Trigonometric Functions

$\sin x$
$\cos x$
$\tan x$
$\cot x$
$\sec x$
$\csc x$

Logarithmic and Exponential Functions

Logarithmic and exponential functions are inverses of each other: \begin{eqnarray*} y=\log_b x & \quad{\small\textrm{if and only if}} & x=b^y\\ y=\ln x & {\small\textrm{ if and only if }} & x=e^y. \end{eqnarray*} In words, $\displaystyle \log_b x$ is the exponent you put on base $b$ to get $x$. Thus, \[log_b b^x=x \qquad {\small\textrm{and}} \qquad b^{\log_b x}=x.\]

More Properties of Logarithmic and Exponential Functions

Notice the relationship between each pair of identities: \[\begin{array}{ccc@{\qquad}ccc} \log_b 1=0 & \longleftrightarrow & b^0=1 & \log_b ac=\log_b a+\log_b c & \longleftrightarrow & b^mb^n=b^{m+n}\\ \log_b b=1 & \longleftrightarrow & b^1=b & \log_b \displaystyle\frac{a}{c}=\log_b a-\log_b c & \longleftrightarrow & \displaystyle\frac{b^m}{b^n}=b^{m-n}\\ \log_b \displaystyle\frac{1}{c}=-\log_b c & \longleftrightarrow & b^{-m}=\displaystyle\frac{1}{b^m} & \log_b a^r=r\log_b a & \longleftrightarrow & (b^m)^n=b^{mn} . \end{array}\]

Graphs of Logarithmic and Exponential Functions

Notice that each curve is the reflection of the other about the line $y=x$.
$f(x) = \ln x$
$f(x) = e^x$

Limits of Logarithmic and Exponential Functions

  1. $\displaystyle \lim_{x\to\infty} \frac{\ln x}{x}=0\quad$ ($\ln x$ grows more slowly than $x$).

  2. $\displaystyle \lim_{x\to\infty} \frac{e^x}{x^n}=\infty$ for all positive integers $n\quad$ ($\displaystyle e^x$ grows faster than $x^n$).

  3. For $|x|\ll 1$, $\displaystyle\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n=e^x$.