Trig, Exponentials, Logs, and Inverses: Foundations for Calculus

Trigonometric Functions and the Unit Circle

Unit Circle and Trigonometric Definitions

The unit circle is a circle of radius 1 centered at the origin in the coordinate plane. It is fundamental in defining the trigonometric functions for all real numbers.

• Coordinates: Any point on the unit circle can be represented as $(x, y)$, where $x = \cos(\theta)$ and $y = \sin(\theta)$ for an angle $\theta$ measured from the positive $x$-axis.

• Trigonometric Ratios: For a general circle of radius $r$ and a point $(x, y)$ on the circle, the trigonometric functions are defined as:

• $\sin(\theta) = \dfrac{y}{r}$

• $\cos(\theta) = \dfrac{x}{r}$

• $\tan(\theta) = \dfrac{y}{x}$

• $\csc(\theta) = \dfrac{r}{y}$

• $\sec(\theta) = \dfrac{r}{x}$

• $\cot(\theta) = \dfrac{x}{y}$

Example: On the unit circle, $r = 1$, so $\sin(\theta) = y$ and $\cos(\theta) = x$.

Common Angles and Values

Trigonometric functions have well-known values at standard angles. These are often summarized in a table for quick reference.

Angle (degrees)	Angle (radians)	$\sin(\theta)$	$\cos(\theta)$	$\tan(\theta)$
0	$0$	$0$	$1$	$0$
30	$\dfrac{\pi}{6}$	$\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{\sqrt{3}}$
45	$\dfrac{\pi}{4}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{2}}{2}$	$1$
60	$\dfrac{\pi}{3}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$	$\sqrt{3}$
90	$\dfrac{\pi}{2}$	$1$	$0$	undefined

Additional info: Table values are representative; more angles can be added for completeness.

Graphs of Trigonometric Functions

The graphs of the six trigonometric functions ($\sin x$, $\cos x$, $\tan x$, $\csc x$, $\sec x$, $\cot x$) are periodic and have characteristic shapes:

• $\sin x$ and $\cos x$: Amplitude 1, period $2\pi$.

• $\tan x$ and $\cot x$: Period $\pi$, with vertical asymptotes where the function is undefined.

• $\sec x$ and $\csc x$: Reciprocal functions of $\cos x$ and $\sin x$ respectively, with vertical asymptotes at zeros of the denominator.

Example: The graph of $y = \sin x$ oscillates between $-1$ and $1$with period $2\pi$.

Transformations of Functions

Shifts, Stretches, and Reflections

Functions can be transformed by shifting, stretching, compressing, and reflecting. For trigonometric functions, a general transformation is:

$f(x) = A \sin(B(x - C)) + D$

• Amplitude ($|A|$): The height from the centerline to the peak. If $A$ is negative, the graph is reflected over the $x$-axis.

• Period ($\dfrac{2\pi}{|B|}$): The length of one cycle. If $B$ is negative, the graph is reflected over the $y$-axis.

• Horizontal Shift ($C$): Moves the graph left or right.

• Vertical Shift ($D$): Moves the graph up or down.

Example: $f(x) = 2\sin(3(x - \pi)) - 1$ has amplitude 2, period $\dfrac{2\pi}{3}$, shifted right by $\pi$, and down by 1.

Exponential Functions

Definition and Properties

An exponential function has the form:

$f(x) = a^x, \quad a > 0, \ a \neq 1$

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• Growth/Decay: If $a > 1$, the function grows; if $0 < a < 1$, it decays.

Example: $f(x) = 2^x$ is an increasing exponential function.

Logarithmic Functions

Definition and Properties

A logarithmic function is the inverse of an exponential function and has the form:

$f(x) = \log_a(x), \quad a > 0, \ a \neq 1$

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• Inverse Relationship: $y = a^x \iff x = \log_a(y)$

Example: $f(x) = \log_2(x)$ is the inverse of $g(x) = 2^x$.

Exponential and Logarithmic Rules

There are several important rules for manipulating exponents and logarithms:

Exponent Rule	Log Rule
$a^{x+y} = a^x \cdot a^y$	$\log_a(xy) = \log_a(x) + \log_a(y)$
$a^{x-y} = \dfrac{a^x}{a^y}$	$\log_a\left(\dfrac{x}{y}\right) = \log_a(x) - \log_a(y)$
$(a^x)^y = a^{xy}$	$\log_a(x^y) = y \cdot \log_a(x)$
$a^1 = a$	$\log_a(a) = 1$
$a^0 = 1$	$\log_a(1) = 0$

One-to-One and Invertible Functions

One-to-One Functions

A function is one-to-one if each output is produced by exactly one input. Formally:

• If $f(x_1) = f(x_2)$, then $x_1 = x_2$.

• If $x_1 \neq x_2$, then $f(x_1) \neq f(x_2)$.

• No two different inputs have the same output.

Example: $f(x) = x^3$ is one-to-one, but $f(x) = x^2$ is not.

Invertible Functions and the Horizontal Line Test

A function is invertible if and only if it is one-to-one. Invertible functions pass the horizontal line test: any horizontal line crosses the graph at most once.

• Inverse Functions: If $f(x)$ is invertible, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• Example: $f(x) = a^x$ and $g(x) = \log_a(x)$ are inverses of each other.

Finding Inverses: To find the inverse of a function, solve for $x$ in terms of $y$, then interchange $x$ and $y$.

Inverse Trigonometric Functions

Definition and Restrictions

Since trigonometric functions are not one-to-one over their entire domains, their inverses are defined by restricting the domain:

• arcsin(y): The value $x$ in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ such that $y = \sin(x)$, for $-1 \leq y \leq 1$.

• arccos(y): The value $x$ in $[0, \pi]$ such that $y = \cos(x)$, for $-1 \leq y \leq 1$.

• arctan(y): The value $x$ in $(-\frac{\pi}{2}, \frac{\pi}{2})$ such that $y = \tan(x)$, for all real $y$.

Example: $\arcsin(\frac{1}{2}) = \frac{\pi}{6}$ because $\sin(\frac{\pi}{6}) = \frac{1}{2}$.