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. This tutorial follows and is a derivative of the one found in HMC Mathematics Online Tutorial.

Review of Trigonometric Functions


Trigonometric functions have an angle for the argument. Before we discuss the function we need to refresh out knowledge on how the angles are measured. There are two ways to measure angles: using degrees, or using radians.

The original motivation for choosing the degree as a unit of rotations and angles is unknown. A degree is a measurement of plane angle, representing $1/360$ of a full rotation. Thus the full rotation is $360^{\circ}$. Half of the full rotation is $180^{\circ}$, one forth is $90^{\circ}$ and so on.


Radian is much more natural way to measure angles. Radian is the ratio between the length of an arc and its radius. Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, $\theta= s /r$, where $\theta$ is the subtended angle in radians, $s$ is arc length, and $r$ is radius. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, $s = r\theta$. A complete revolution is $2\pi$ radians (as shown with a circle of radius one and circumference $2\pi$).

It follows that the magnitude in radians of one complete revolution ($360^{\circ}$ degrees) is the length of the entire circumference divided by the radius, or $2\pi r /r$, or $2\pi$. Thus $2\pi$ radians is equal to $360^{\circ}$ degrees, meaning that one radian is equal to $180/\pi\approx57.3^{\circ}$ degrees.

$\begin{array}{l} {\small\textrm{Measure }} \theta {\small\textrm{ in radians:}}\\ \theta =\frac{{\small\textrm{arc length}}}{{\small\textrm{radius}}}\\ ~\\ 180^{\circ}=\displaystyle\frac{\pi r}{r}=\pi {\small\textrm{ radians}}\\ ~\\ {\small\textrm{Radians}}=\displaystyle\frac{{\small\textrm{degrees}}}{180}\cdot \pi \end{array}$

Geometrically, there are two ways to describe trigonometric functions:

Polar Angle

Unit circle
$\begin{array}{l} x=\cos\theta\\ y=\sin\theta\\ \end{array}$

Right Triangle

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

$\theta$ in rad. $0 {\small\textrm{ rad}} $ $\pi/6 {\small\textrm{ rad}} $ $\pi/4 {\small\textrm{ rad}} $ $\pi/3 {\small\textrm{ rad}} $ $\pi/2 {\small\textrm{ rad}}$
$\theta$ in deg. $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} $ ${\small\textrm{undefined}}$
Figure 1
Figure 1

Trigonometric Identities

We list here some of the most commonly used identities:
  • $\sin(-\theta) = -\sin\theta $
  • $\cos(-\theta)= \cos\theta $
  • $\cos(\theta+\pi) = -\cos\theta $
  • $\sin(\theta+\pi) = -\sin\theta $
  • $\sin(\theta +\pi/2) = \cos\theta$
  • $\cos(\theta +\pi/2) = -\sin\theta$
  • $\cos(\theta +2\pi) = \cos\theta$
  • $\sin(\theta +2\pi) = \sin\theta$
  • $\cos^2\theta+\sin^2\theta=1 $
  • $\cos^2\theta =\displaystyle\frac{1}{2}[1+\cos(2\theta)] $
  • $\sin^2\theta=\displaystyle\frac{1}{2}[1-\cos(2\theta)]$
  • $\sin(2\theta)=2\sin\theta\cos\theta $
  • $\cos(2\theta)=\cos^2\theta-\sin^2\theta $
  • $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$
  • $\cos(\alpha +\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$
  • $C_1\cos(\omega x)+C_2\sin(\omega x)=A\sin(\omega x+\phi) \mbox{, where } A=\sqrt{C_1^2+C_2^2},\quad \phi=\arctan (C_1/C_2)$

Graphs of Trigonometric Functions

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

Review of 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

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

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