Precalculus Study Guide: Trigonometric Functions, Identities, and Applications

Trigonometric Inverse Functions

Ranges of Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when the value of a trigonometric function is known. Each inverse function has a specific range where it produces principal values:

• $ext{sin}^{-1} x$: $[- rac{\pi}{2}, rac{\pi}{2}]$

• $ext{cos}^{-1} x$: $[0, \pi]$

• $ext{tan}^{-1} x$: $[- rac{\pi}{2}, rac{\pi}{2}]$

• $ext{cot}^{-1} x$: $(0, \\pi)$

• $ext{csc}^{-1} x$: $[- rac{\pi}{2}, 0) \\cup (0, rac{\pi}{2}]$

• $ext{sec}^{-1} x$: $[0, rac{\pi}{2}) \\cup ( rac{\pi}{2}, \\pi]$

Example: $\theta = ext{arccos}(-\frac{\sqrt{2}}{2})$ yields $\theta = \frac{3\pi}{4}$.

Exact Values and Reference Triangles

Finding Exact Values Without a Calculator

Reference triangles help determine the exact values of trigonometric expressions, especially when the angle is in a specific quadrant.

• arccos and arcsin functions return principal values within their defined ranges.

• Use the unit circle and special triangles (30-60-90, 45-45-90) for exact values.

Example: $\theta = \text{arccos}(-\frac{\sqrt{2}}{2})$ corresponds to $\theta = \frac{3\pi}{4}$.

Algebraic Expressions Using Inverse Functions

Expressing Trigonometric Functions Algebraically

Some problems require expressing trigonometric functions in terms of $x$ without using inverse or trigonometric functions.

• Use right triangle relationships and Pythagorean identities.

• For $\sin(\arccot(x))$, draw a triangle where $\cot(\theta) = x$ and use $\sin(\theta) = \frac{1}{\sqrt{1+x^2}}$.

Solving Trigonometric Equations

Finding All Solutions in a Given Interval

Trigonometric equations often have multiple solutions within a specified interval.

• Set the equation to zero and solve for $x$ using algebraic manipulation and identities.

• List all solutions in the interval $-2\pi \leq x \leq 2\pi$.

Example: $\tan^2 x - \sqrt{3} \tan x = 0$ yields $\tan x = 0$ or $\tan x = \sqrt{3}$.

Graphing Trigonometric Functions

Amplitude, Period, Vertical Shift, and Phase Shift

For a function $f(x) = a \sin(bx + c) + d$:

• Amplitude: $|a|$

• Period: $\frac{2\pi}{|b|}$

• Vertical Shift: $d$

• Phase Shift: $-\frac{c}{b}$

Example: For $f(x) = 2 \sin(\frac{\pi}{4}x + \frac{\pi}{3}) + 1$:

• Amplitude: 2

• Period: $\frac{2\pi}{\frac{\pi}{4}} = 8$

• Vertical Shift: 1

• Phase Shift: $-\frac{\frac{\pi}{3}}{\frac{\pi}{4}} = -\frac{4}{3}$

Sum and Difference Formulas

Using Trigonometric Identities

Sum and difference identities allow calculation of trigonometric values for sums or differences of angles:

• $\sin(x + y) = \sin x \cos y + \cos x \sin y$

• $\cos(x + y) = \cos x \cos y - \sin x \sin y$

Example: Given $\sin x = -\frac{4}{5}$ (Quadrant III) and $\cos y = \frac{\sqrt{2}}{2}$ (Quadrant IV), find $\sin(x + y)$ using the identity above.

Trigonometric Equations and Identities

Verifying and Using Identities

Trigonometric identities are equations that are true for all values in the domain. They are used to simplify expressions and solve equations.

• Pythagorean Identity: $\sin^2 x + \cos^2 x = 1$

• Double Angle Identity: $\cos(2x) = \cos^2 x - \sin^2 x$

• Sum-to-Product and Product-to-Sum: Useful for simplifying products and sums of sines and cosines.

Sample Table: Common Trigonometric Identities

Identity	Equation
Pythagorean	$\sin^2 x + \cos^2 x = 1$
Double Angle	$\cos(2x) = \cos^2 x - \sin^2 x$
Sum Formula	$\sin(x + y) = \sin x \cos y + \cos x \sin y$
Difference Formula	$\sin(x - y) = \sin x \cos y - \cos x \sin y$

Applications of Trigonometry

Solving Real-World Problems

Trigonometry is used to solve problems involving angles and distances, such as finding the distance between two objects using angle of depression or elevation.

• Draw diagrams to represent the situation.

• Use trigonometric ratios (sine, cosine, tangent) to relate sides and angles.

Example: Given two towers and angles of depression, use the Law of Sines or Law of Cosines to find the distance between them.

Graphing Inverse Trigonometric Functions

Plotting $\sin(x)$ and $\sin^{-1}(x)$

Inverse trigonometric functions have restricted domains and ranges. When graphing $\sin(x)$ and $\sin^{-1}(x)$:

• $\sin(x)$: Domain $(-\infty, \infty)$, Range $[-1, 1]$

• $\sin^{-1}(x)$: Domain $[-1, 1]$, Range $[-\frac{\pi}{2}, \frac{\pi}{2}]$

• Label key points such as $(0, 0)$, $(\frac{\pi}{2}, 1)$, $(-\frac{\pi}{2}, -1)$ for $\sin(x)$ and $(-1, -\frac{\pi}{2})$, $(0, 0)$, $(1, \frac{\pi}{2})$ for $\sin^{-1}(x)$.

Trigonometric Proofs and Identities

Proving Trigonometric Identities

To prove identities, manipulate one side using algebraic and trigonometric properties until it matches the other side.

• Use fundamental identities and algebraic manipulation.

• Example: $\cos(2x) = \cos^2 x - \sin^2 x$

Triangles and Trigonometric Ratios

Using Right Triangles

Right triangles are used to define trigonometric ratios and solve for unknown sides or angles.

• For a triangle with sides 2 cm and 4 cm, use Pythagoras' theorem to find the hypotenuse and trigonometric ratios.

Even and Odd Functions

Properties of Cosine and Sine

Cosine is an even function, meaning $\cos(-x) = \cos(x)$. Sine is an odd function, meaning $\sin(-x) = -\sin(x)$.

• Use identities to prove these properties.

Summary Table: Key Trigonometric Properties

Function	Even/Odd	Identity
$\sin(x)$	Odd	$\sin(-x) = -\sin(x)$
$\cos(x)$	Even	$\cos(-x) = \cos(x)$

Additional info:

• Some problems may require drawing reference triangles or using the unit circle for visualization.

• For graphing, always label axes and key points for clarity.

• When solving equations, check all possible solutions within the given interval.