Precalculus Study Guide: Trigonometric Functions, Identities, and Applications

Trigonometric Inverse Functions and Their Ranges

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when given trigonometric ratios. Each function has a specific range to ensure it is one-to-one and thus invertible.

• arcsin(x): The inverse sine function, with range $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$

• arccos(x): The inverse cosine function, with range $[0, \pi]$

• arctan(x): The inverse tangent function, with range $\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$

• arccsc(x): The inverse cosecant function, with range $\left[ -\frac{\pi}{2}, 0 \right) \cup \left( 0, \frac{\pi}{2} \right]$

• arcsec(x): The inverse secant function, with range $[0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi]$

• arccot(x): The inverse cotangent function, with range $(0, \pi)$

Evaluating Inverse Trigonometric Expressions

Finding Exact Values

To find the exact value of an inverse trigonometric expression, it is often helpful to draw a reference triangle and consider the function's range.

• Example: $\arccos\left(-\frac{1}{2}\right)$

  • Cosine is negative in quadrants II and III, but the range of arccos is $[0, \pi]$, so the answer is $\frac{2\pi}{3}$.

• Example: $\csc^{-1}(-2)$

  • Cosecant is negative in quadrants III and IV, but the range of arccsc is $\left[ -\frac{\pi}{2}, 0 \right) \cup \left( 0, \frac{\pi}{2} \right]$; the answer is $-\frac{\pi}{6}$.

Compositions of Trigonometric and Inverse Trigonometric Functions

Evaluating Compositions

When evaluating expressions like $\sin(\arccos(x))$, use a reference triangle to relate the functions.

• Example: $\sin(\arccos(x))$

  • Let $\theta = \arccos(x)$, so $\cos(\theta) = x$.

  • In a right triangle, adjacent/hypotenuse = $x/1$; thus, opposite = $\sqrt{1 - x^2}$.

  • Therefore, $\sin(\arccos(x)) = \sqrt{1 - x^2}$.

Solving Trigonometric Equations

Solving for All Solutions in a Given Interval

To solve trigonometric equations, use algebraic manipulation and knowledge of the unit circle.

• Example: $\tan^2 x - \sqrt{3} \tan x = 0$

  • Factor: $\tan x (\tan x - \sqrt{3}) = 0$

  • Solutions: $\tan x = 0 \implies x = 0, \pi, 2\pi, ...$; $\tan x = \sqrt{3} \implies x = \frac{\pi}{3}, \frac{4\pi}{3}, ...$

• Example: $\sqrt{2} \sin x + 1 = 0$

  • $\sin x = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}$

  • Solutions in $[-2\pi, 2\pi]$: $x = -\frac{3\pi}{4}, -\frac{7\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$

Graphing Trigonometric Functions

Amplitude, Period, Phase Shift, and Vertical Shift

The general form of a sine function is $f(x) = a \sin(bx + c) + d$.

• Amplitude: $|a|$

• Period: $\frac{2\pi}{|b|}$

• Phase Shift: $-\frac{c}{b}$

• Vertical Shift: $d$

Example: For $f(x) = 2 \sin\left(\frac{2\pi}{5}x + \frac{\pi}{5}\right) + 1$:

• Amplitude: $2$

• Period: $5$

• Vertical Shift: $1$

• Phase Shift: $-\frac{\pi/5}{2\pi/5} = -\frac{1}{2}$

Sum and Difference Formulas

Using Identities to Find Exact Values

Sum and difference formulas allow us to compute trigonometric values for sums or differences of angles.

• Sine: $\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$

• Cosine: $\cos(x \pm y) = \cos x \cos y \mp \sin x \sin y$

Example: Find $\sin(75^\circ)$:

• $\sin(75^\circ) = \sin(30^\circ + 45^\circ) = \sin 30^\circ \cos 45^\circ + \cos 30^\circ \sin 45^\circ$

• Plug in values: $\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4}$

Applications of Trigonometry

Angle of Depression and Distance Problems

Trigonometric functions are used to solve real-world problems involving angles of elevation and depression.

• Example: A pilot at an altitude of 2.7 km sights two control towers. The angle of depression to the closest tower is $58^\circ$, and to the farther tower is $37^\circ$. Find the distance between the towers.

• Let $y$ be the horizontal distance to the closer tower, $x$ the distance between towers.

• $\tan 58^\circ = \frac{2.7}{y}$, $\tan 37^\circ = \frac{2.7}{x + y}$

• Solve for $x$: $x = y - (x + y) = 2.7 \tan 53^\circ - 2.7 \tan 32^\circ = 1.896$ km

Trigonometric Identities and Verifications

Verifying Identities

Trigonometric identities are equations that are true for all values in the domain. Verifying identities involves algebraic manipulation to show both sides are equal.

• Example: $1 - \cos^2 x = \sin^2 x$ (Pythagorean Identity)

• Example: $\cos^2 x = 1 - \sin^2 x$

• Example: $\sec x - \cot x \cdot \cos x = \frac{1}{\csc x}$

Tables: Trigonometric Function Ranges and Values

Ranges of Inverse Trigonometric Functions

Function	Range
$\sin^{-1} x$	$\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$
$\cos^{-1} x$	$[0, \pi]$
$\tan^{-1} x$	$\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$
$\csc^{-1} x$	$\left[ -\frac{\pi}{2}, 0 \right) \cup \left( 0, \frac{\pi}{2} \right]$
$\sec^{-1} x$	$[0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi]$
$\cot^{-1} x$	$(0, \pi)$

Summary of Key Concepts

• Inverse trigonometric functions have restricted ranges to ensure they are functions.

• Reference triangles are useful for evaluating compositions of trigonometric and inverse trigonometric functions.

• Sum and difference identities allow for the calculation of trigonometric values for non-standard angles.

• Trigonometric equations can be solved algebraically and graphically.

• Trigonometric identities are essential for simplifying expressions and verifying equalities.

• Applications include solving real-world problems involving angles of elevation and depression.