Trigonometric Functions: Definitions, Properties, and Applications

Trigonometric Functions and Their Definitions

Right Triangle Definitions

Trigonometric functions are initially defined using right triangles, relating the angles to the ratios of side lengths.

• Sine (sin): $\sin(\theta) = \dfrac{\text{opposite}}{\text{hypotenuse}}$

• Cosine (cos): $\cos(\theta) = \dfrac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent (tan): $\tan(\theta) = \dfrac{\text{opposite}}{\text{adjacent}}$

• Cosecant (csc): $\csc(\theta) = \dfrac{\text{hypotenuse}}{\text{opposite}}$

• Secant (sec): $\sec(\theta) = \dfrac{\text{hypotenuse}}{\text{adjacent}}$

• Cotangent (cot): $\cot(\theta) = \dfrac{\text{adjacent}}{\text{opposite}}$

These definitions apply when the triangle fits within a quadrant, as shown in the diagrams.

Unit Circle Definitions

To generalize trigonometric functions for all real numbers, we use the unit circle (a circle of radius 1 centered at the origin).

• Sine (sin): $\sin(\theta) = y$ (y-coordinate on the unit circle)

• Cosine (cos): $\cos(\theta) = x$ (x-coordinate on the unit circle)

• Tangent (tan): $\tan(\theta) = \dfrac{y}{x}$

• Cosecant (csc): $\csc(\theta) = \dfrac{1}{y}$

• Secant (sec): $\sec(\theta) = \dfrac{1}{x}$

• Cotangent (cot): $\cot(\theta) = \dfrac{x}{y}$

These definitions allow us to evaluate trigonometric functions for any angle, not just those in right triangles.

Expanded Definitions for Any Circle Centered at the Origin

Trigonometric functions can be defined for any circle of radius r centered at the origin:

• $\sin(\theta) = \dfrac{y}{r}$

• $\cos(\theta) = \dfrac{x}{r}$

• $\tan(\theta) = \dfrac{y}{x}$

• $\csc(\theta) = \dfrac{r}{y}$

• $\sec(\theta) = \dfrac{r}{x}$

• $\cot(\theta) = \dfrac{x}{y}$

Evaluating Trigonometric Functions

Using the Unit Circle

To evaluate trigonometric functions at specific angles, locate the corresponding point on the unit circle.

• Example: $\theta = 90^\circ$ corresponds to the point (0,1) on the unit circle.

• $\sin(90^\circ) = 1$

• $\cos(90^\circ) = 0$

• $\tan(90^\circ)$ is undefined

• $\csc(90^\circ) = 1$

• $\sec(90^\circ)$ is undefined

• $\cot(90^\circ) = 0$

Reference Angles and Quadrants

Reference angles help evaluate trigonometric functions for angles outside the first quadrant. The sign of the function depends on the quadrant.

Quadrant	Positive Functions	Negative Functions
I	Sine, Cosine, Tangent	None
II	Sine, Cosecant	Cosine, Secant, Tangent, Cotangent
III	Tangent, Cotangent	Sine, Cosine, Secant, Cosecant
IV	Cosine, Secant	Sine, Tangent, Cosecant, Cotangent

Negative Angles

Trigonometric functions of negative angles have specific identities:

Function	Identity
$\sin(-\theta)$	$-\sin(\theta)$
$\cos(-\theta)$	$\cos(\theta)$
$\tan(-\theta)$	$-\tan(\theta)$
$\csc(-\theta)$	$-\csc(\theta)$
$\sec(-\theta)$	$\sec(\theta)$
$\cot(-\theta)$	$-\cot(\theta)$

Trigonometric Identities

Pythagorean Identities

The Pythagorean identities are fundamental relationships among trigonometric functions:

• $\sin^2(\theta) + \cos^2(\theta) = 1$

• $1 + \tan^2(\theta) = \sec^2(\theta)$

• $1 + \cot^2(\theta) = \csc^2(\theta)$

Solving Trigonometric Equations

General Approach

To solve trigonometric equations, isolate the trigonometric function and use reference angles and the unit circle to find all solutions within the given interval.

• Example: Solve $\cos(\theta) = -\dfrac{\sqrt{2}}{2}$ for $0 \leq \theta < 2\pi$.

• Reference angle: $\dfrac{\pi}{4}$

• Solutions: $\theta = \dfrac{3\pi}{4}, \dfrac{5\pi}{4}$

Solving with Inverse Functions

For equations like $\cos(\theta) = a$, use $\theta = \arccos(a)$ and consider all quadrants where cosine has the same sign.

Applications: Arc Length and Displacement

Arc Length

The arc length $s$ of a circle of radius $r$ subtended by an angle $\theta$ (in radians) is:

• $s = r\theta$

Displacement on the Unit Circle

To find the distance between two points on the unit circle corresponding to angles $\theta_1$ and $\theta_2$:

• Let the points be $(\cos(\theta_1), \sin(\theta_1))$ and $(\cos(\theta_2), \sin(\theta_2))$

• Distance: $d = \sqrt{[\cos(\theta_2) - \cos(\theta_1)]^2 + [\sin(\theta_2) - \sin(\theta_1)]^2}$

Summary Table: Trigonometric Functions (Any Circle Centered at the Origin)

Function	Definition
$\sin(\theta)$	$\dfrac{y}{r}$
$\cos(\theta)$	$\dfrac{x}{r}$
$\tan(\theta)$	$\dfrac{y}{x}$
$\csc(\theta)$	$\dfrac{r}{y}$
$\sec(\theta)$	$\dfrac{r}{x}$
$\cot(\theta)$	$\dfrac{x}{y}$

Additional info:

• These notes include both the geometric and algebraic perspectives on trigonometric functions, as well as their applications to solving equations and real-world problems involving circles.