Trigonometric Functions, Properties, Graphs, Inverses, and Identities: Precalculus Study Guide

Angles and Their Measurement

Degrees and Radians

Angles are measured by the amount of rotation between two rays sharing a common origin. The initial ray is called the initial side, and the other is the terminal side. Angles are typically measured in the counter-clockwise direction.

• Degree: A circle is divided into 360 degrees. One degree ($1^\circ$) is $\frac{1}{360}$ of a full rotation.

• Radian: The radian is based on the arc length of a circle. One radian is the angle subtended by an arc equal in length to the radius.

• Conversion:

  • Degrees to radians: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$

  • Radians to degrees: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$

Example: Convert $30^\circ$ to radians:

$30^\circ \times \frac{\pi}{180} = \frac{\pi}{6}$

Example: Convert $\frac{\pi}{4}$ to degrees:

$\frac{\pi}{4} \times \frac{180}{\pi} = 45^\circ$

Arc Length and Area of a Sector

For a circle of radius $r$ and central angle $\theta$ (in radians):

• Arc length: $s = r\theta$

• Area of sector: $A = \frac{1}{2} \theta r^2$

Example: Find the arc length for $r = 2$, $\theta = 90^\circ$:

First, convert $90^\circ$ to radians: $90^\circ \times \frac{\pi}{180} = \frac{\pi}{2}$

Then, $s = 2 \times \frac{\pi}{2} = \pi$

Trigonometric Functions

Definitions on the Unit Circle

The unit circle (radius 1, centered at the origin) is fundamental for defining trigonometric functions. For a point $P = (x, y)$ on the unit circle corresponding to angle $\theta$:

• Sine: $\sin \theta = y$

• Cosine: $\cos \theta = x$

• Tangent: $\tan \theta = \frac{y}{x}$

• Cotangent: $\cot \theta = \frac{x}{y}$

• Cosecant: $\csc \theta = \frac{1}{y}$

• Secant: $\sec \theta = \frac{1}{x}$

Example: For $\theta = \frac{\pi}{6}$, the unit circle coordinates are $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, so $\sin \frac{\pi}{6} = \frac{1}{2}$, $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$.

Exact Values at Key Angles

Common angles and their coordinates on the unit circle:

• $0^\circ$ ($0$):$(1, 0)$

• $30^\circ$ ($\frac{\pi}{6}$): $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

• $45^\circ$ ($\frac{\pi}{4}$): $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$

• $60^\circ$ ($\frac{\pi}{3}$): $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$

• $90^\circ$ ($\frac{\pi}{2}$): $(0, 1)$

• $180^\circ$ ($\pi$): $(-1, 0)$

• $270^\circ$ ($\frac{3\pi}{2}$): $(0, -1)$

• $360^\circ$ ($2\pi$): $(1, 0)$

Example: $\sin \frac{\pi}{6} = \frac{1}{2}$, $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$

Properties of Trigonometric Functions

Domains and Ranges

The domain and range of each trigonometric function depend on their definitions and the unit circle.

Function	Domain	Range
sin $\theta$	All real numbers	$[-1, 1]$
cos $\theta$	All real numbers	$[-1, 1]$
tan $\theta$	$\theta \neq k\frac{\pi}{2}$, $k$ odd integer	All real numbers
cot $\theta$	$\theta \neq k\pi$, $k$ integer	All real numbers
sec $\theta$	$\theta \neq k\frac{\pi}{2}$, $k$ odd integer	$(-\infty, -1] \cup [1, \infty)$
csc $\theta$	$\theta \neq k\pi$, $k$ integer	$(-\infty, -1] \cup [1, \infty)$

Periodicity

• $\sin(\theta + 2\pi) = \sin \theta$

• $\cos(\theta + 2\pi) = \cos \theta$

• $\tan(\theta + \pi) = \tan \theta$

• $\cot(\theta + \pi) = \cot \theta$

• $\sec(\theta + 2\pi) = \sec \theta$

• $\csc(\theta + 2\pi) = \csc \theta$

Even and Odd Properties

• $\cos(-\theta) = \cos \theta$ (even)

• $\sin(-\theta) = -\sin \theta$ (odd)

• $\tan(-\theta) = -\tan \theta$ (odd)

• $\cot(-\theta) = -\cot \theta$ (odd)

• $\sec(-\theta) = \sec \theta$ (even)

• $\csc(-\theta) = -\csc \theta$ (odd)

Pythagorean Identities

• $\cos^2 \theta + \sin^2 \theta = 1$

• $\tan^2 \theta + 1 = \sec^2 \theta$

• $\cot^2 \theta + 1 = \csc^2 \theta$

Graphs of Trigonometric Functions

Basic Sine and Cosine Graphs

The graphs of $y = \sin x$ and $y = \cos x$ are periodic and oscillate between $-1$ and $1$.

• Amplitude: $|A|$ in $y = A \sin(\omega x)$ or $y = A \cos(\omega x)$

• Period: $t = \frac{2\pi}{\omega}$

• Phase Shift: $\frac{\phi}{\omega}$ in $y = A \sin(\omega x + \phi)$ or $y = A \cos(\omega x + \phi)$

Graphing Steps:

1. Mark lines $y = \pm A$ (amplitude).

2. Compute the period.

3. Mark key points (e.g., zeros, maxima, minima).

4. Divide the period into four subintervals for plotting.

5. Draw the curve using these points.

Example: $f(x) = 3 \sin(\pi x)$ has amplitude $3$, period$\frac{2\pi}{\pi} = 2$.

Graphs and Properties of Other Trigonometric Functions

• Tangent: Domain excludes odd multiples of $\frac{\pi}{2}$; range is all real numbers; period is $\pi$; odd function.

• Cotangent: Domain excludes integer multiples of $\pi$; range is all real numbers; period is $\pi$; odd function.

• Cosecant: Domain excludes integer multiples of $\pi$; range is $|y| \geq 1$; period is $2\pi$; odd function.

• Secant: Domain excludes odd multiples of $\frac{\pi}{2}$; range is $|y| \geq 1$; period is $2\pi$; even function.

Inverse Trigonometric Functions

Definitions and Domains

Inverse trigonometric functions are defined by restricting the domain of the original function to make it one-to-one.

Function	Domain	Range
$y = \arcsin x$	$-1 \leq x \leq 1$	$-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$
$y = \arccos x$	$-1 \leq x \leq 1$	$0 \leq y \leq \pi$
$y = \arctan x$	All real numbers	$-\frac{\pi}{2} < y < \frac{\pi}{2}$
$y = \arccsc x$	$|x| \geq 1$	$-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}, y \neq 0$
$y = \arcsec x$	$|x| \geq 1$	$0 \leq y \leq \pi, y \neq \frac{\pi}{2}$
$y = \arccot x$	All real numbers	$0 < y < \pi$

Example: $\arccos(\frac{1}{2}) = \frac{\pi}{3}$

Trigonometric Equations and Identities

Solving Trigonometric Equations

To solve equations involving trigonometric functions, use algebraic manipulation, identities, and inverse functions.

• Example: Solve $\tan^2 \theta = 3$ for $0 \leq \theta < 2\pi$.

  • $\tan \theta = \sqrt{3}$ or $\tan \theta = -\sqrt{3}$

  • Find all solutions in the given interval.

• General Solution: For $\tan \theta = a$, $\theta = \arctan a + k\pi$, $k$ integer.

Establishing Trigonometric Identities

A trigonometric identity is an equality that holds for all values in the domain. Common strategies include:

1. Start with the more complex side.

2. Use known identities to simplify.

3. Convert all terms to sines and cosines if possible.

4. Use common denominators for fractions.

Common Identities:

• $\tan x = \frac{\sin x}{\cos x}$

• $\sec x = \frac{1}{\cos x}$

• $\cot x = \frac{\cos x}{\sin x}$

• $\csc x = \frac{1}{\sin x}$

• $\cos(-x) = \cos x$

• $\sin(-x) = -\sin x$

• $1 = \cos^2 x + \sin^2 x$

• $\sec^2 x = 1 + \tan^2 x$

Example: Prove $\cos x \cot x - \csc x = -\sin x$

Rewrite $\cot x = \frac{\cos x}{\sin x}$, $\csc x = \frac{1}{\sin x}$:

$\cos x \cot x - \csc x = \cos x \left(\frac{\cos x}{\sin x}\right) - \frac{1}{\sin x} = \frac{\cos^2 x - 1}{\sin x} = \frac{-\sin^2 x}{\sin x} = -\sin x$

Summary Table: Trigonometric Functions

Function	Domain	Range	Period	Even/Odd
$\sin x$	All real numbers	$[-1, 1]$	$2\pi$	Odd
$\cos x$	All real numbers	$[-1, 1]$	$2\pi$	Even
$\tan x$	$x \neq k\frac{\pi}{2}$, $k$ odd	All real numbers	$\pi$	Odd
$\cot x$	$x \neq k\pi$, $k$ integer	All real numbers	$\pi$	Odd
$\sec x$	$x \neq k\frac{\pi}{2}$, $k$ odd	$(-\infty, -1] \cup [1, \infty)$	$2\pi$	Even
$\csc x$	$x \neq k\pi$, $k$ integer	$(-\infty, -1] \cup [1, \infty)$	$2\pi$	Odd

Additional info: The notes include graphical representations and example problems for each function, as well as step-by-step instructions for graphing trigonometric functions with amplitude, period, and phase shift. The study guide expands on brief points to provide full academic context and formulas.