Precalculus Trigonometry: Functions, Identities, and Applications Study Notes

Trigonometric Functions and Their Inverses

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine angles when given the value of a trigonometric ratio. The principal values are typically restricted to specific intervals to ensure the function is one-to-one.

• arccos(x): The inverse of the cosine function, returns an angle whose cosine is x. Principal values: $0 \leq \theta \leq \pi$.

• arcsin(x): The inverse of the sine function, returns an angle whose sine is x. Principal values: $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$.

• arctan(x): The inverse of the tangent function, returns an angle whose tangent is x. Principal values: $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$.

• arccot(x): The inverse of the cotangent function, returns an angle whose cotangent is x.

• arcsec(x): The inverse of the secant function, returns an angle whose secant is x.

• arccsc(x): The inverse of the cosecant function, returns an angle whose cosecant is x.

Example: Find $\arccos(0.5)$.

• $\arccos(0.5) = \frac{\pi}{3}$

Evaluating Inverse Trigonometric Expressions

To evaluate expressions like $\sin(\sec^{-1}(-\sqrt{2}))$, first find the angle whose secant is $-\sqrt{2}$, then compute its sine.

• Step 1: Let $\theta = \sec^{-1}(-\sqrt{2})$ so $\sec \theta = -\sqrt{2}$.

• Step 2: $\cos \theta = -\frac{1}{\sqrt{2}}$

• Step 3: $\sin \theta = \pm \sqrt{1 - \cos^2 \theta}$

Trigonometric Identities and Simplification

Basic Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values in their domains.

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

• Quotient Identities: $\tan x = \frac{\sin x}{\cos x}$, $\cot x = \frac{\cos x}{\sin x}$

• Reciprocal Identities: $\sec x = \frac{1}{\cos x}$, $\csc x = \frac{1}{\sin x}$

Simplifying Trigonometric Expressions

Use identities to rewrite and simplify expressions.

• Example: $\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}$

• Combine over a common denominator: $\frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x}$

Trigonometric Functions: Odd, Even, or Neither

Classification of Functions

A function is even if $f(-x) = f(x)$, odd if $f(-x) = -f(x)$, and neither if it satisfies neither property.

• Example: $f(x) = \sin x + \tan x$

• $f(-x) = \sin(-x) + \tan(-x) = -\sin x - \tan x = -f(x)$, so $f(x)$ is odd.

Solving Right Triangles

Right Triangle Properties

In a right triangle, the relationships between the sides and angles are described by trigonometric ratios.

• Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Example: Given $b = 7$, $c = 14$, find $a$.

• Use Pythagorean Theorem: $a^2 + b^2 = c^2$

• $a^2 + 7^2 = 14^2 \implies a^2 = 196 - 49 = 147 \implies a = \sqrt{147} = 7\sqrt{3}$

Applications of Trigonometry

Angles of Elevation and Depression

Angles of elevation and depression are used to solve real-world problems involving heights and distances.

• Angle of Elevation: The angle formed by the horizontal and the line of sight looking up.

• Angle of Depression: The angle formed by the horizontal and the line of sight looking down.

• Example: From the top of a vertical cliff 80 meters above the ocean, the angle of depression to a marker on the surface is $20^\circ$. Find the horizontal distance to the marker.

• Use $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

• $\tan 20^\circ = \frac{80}{d} \implies d = \frac{80}{\tan 20^\circ}$

Factoring Trigonometric Expressions

Factoring Techniques

Factoring trigonometric expressions often involves recognizing patterns and using identities.

• Example: $5 \cos^2 x \sin^6 x - 12 \cos x \sin^8 x + 4 \sin^8 x$

• Factor out $\sin^6 x$:

• $\sin^6 x (5 \cos^2 x - 12 \cos x \sin^2 x + 4 \sin^2 x)$

Table: Trigonometric Functions and Their Inverses

Function	Inverse	Principal Value Range
$\sin x$	$\arcsin x$	$-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$
$\cos x$	$\arccos x$	$0 \leq x \leq \pi$
$\tan x$	$\arctan x$	$-\frac{\pi}{2} < x < \frac{\pi}{2}$
$\sec x$	$\sec^{-1} x$	$0 \leq x < \frac{\pi}{2}$ or $\pi < x \leq 2\pi$
$\csc x$	$\csc^{-1} x$	$0 < x < \pi$
$\cot x$	$\cot^{-1} x$	$0 < x < \pi$

Additional info:

• Some problems require using a calculator to find decimal approximations for inverse trigonometric functions.

• Right triangle problems may involve using the Law of Sines or Law of Cosines for non-right triangles (not shown in this set).