Trigonometry - Complete Guide with Unit Circle

Table of Contents

  1. Introduction and Real-World Applications
  2. Core Concepts and Right Triangle Trig
  3. The Unit Circle
  4. Trigonometric Functions
  5. Trigonometric Identities
  6. Solving Trig Equations
  7. Applications and Word Problems
  8. Practice Problems
  9. Interactive Visualizations
  10. Related Topics

Introduction and Real-World Applications

Trigonometry is the study of triangles and the relationships between their angles and sides. It extends to periodic phenomena, making it essential for understanding waves, oscillations, and circular motion.

Why Learn Trigonometry?

  • Engineering: Design bridges, analyze forces, electrical circuits
  • Physics: Wave motion, optics, quantum mechanics
  • Navigation: GPS systems, aviation, maritime navigation
  • Music & Sound: Audio engineering, acoustics, synthesis
  • Computer Graphics: 3D rendering, game development, animation
  • Architecture: Structural analysis, sun angles, design

The Word "Trigonometry"

From Greek: "trigonon" (triangle) + "metron" (measure)

Literally means "triangle measuring"

Core Concepts and Right Triangle Trig

Basic Trigonometric Ratios

In a right triangle with angle θ:

Sine (sin)

$$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$$

Cosine (cos)

$$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$

Tangent (tan)

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

Reciprocal Functions

Cosecant (csc)

$$\csc \theta = \frac{1}{\sin \theta}$$

Secant (sec)

$$\sec \theta = \frac{1}{\cos \theta}$$

Cotangent (cot)

$$\cot \theta = \frac{1}{\tan \theta}$$

Special Right Triangles

30-60-90 Triangle

Sides in ratio: $1 : \sqrt{3} : 2$

  • $\sin 30° = \frac{1}{2}$, $\cos 30° = \frac{\sqrt{3}}{2}$, $\tan 30° = \frac{1}{\sqrt{3}}$
  • $\sin 60° = \frac{\sqrt{3}}{2}$, $\cos 60° = \frac{1}{2}$, $\tan 60° = \sqrt{3}$

45-45-90 Triangle

Sides in ratio: $1 : 1 : \sqrt{2}$

  • $\sin 45° = \cos 45° = \frac{\sqrt{2}}{2}$, $\tan 45° = 1$

The Unit Circle

Definition and Properties

The unit circle is a circle with radius 1 centered at the origin (0,0)

Equation: $x^2 + y^2 = 1$

For any angle θ measured from the positive x-axis:

  • $x = \cos \theta$
  • $y = \sin \theta$
  • Point on circle: $(\cos \theta, \sin \theta)$

Key Angles and Values

Degrees Radians sin θ cos θ tan θ
0° 0 0 1 0
30° π/6 1/2 √3/2 1/√3
45° π/4 √2/2 √2/2 1
60° π/3 √3/2 1/2 √3
90° π/2 1 0 undefined

Radian Measure

Definition: One radian is the angle subtended by an arc equal to the radius

Conversions:

  • $180° = \pi$ radians
  • Degrees to radians: multiply by $\frac{\pi}{180}$
  • Radians to degrees: multiply by $\frac{180}{\pi}$

Trigonometric Functions

Graphs of Trig Functions

Sine Function: $y = \sin x$

  • Period: $2\pi$
  • Amplitude: 1
  • Domain: all real numbers
  • Range: $[-1, 1]$
  • Odd function: $\sin(-x) = -\sin(x)$

Cosine Function: $y = \cos x$

  • Period: $2\pi$
  • Amplitude: 1
  • Domain: all real numbers
  • Range: $[-1, 1]$
  • Even function: $\cos(-x) = \cos(x)$

Tangent Function: $y = \tan x$

  • Period: $\pi$
  • Domain: all real numbers except $x = \frac{\pi}{2} + n\pi$
  • Range: all real numbers
  • Odd function: $\tan(-x) = -\tan(x)$
  • Vertical asymptotes at $x = \frac{\pi}{2} + n\pi$

Transformations of Trig Functions

General form: $y = A \sin(B(x - C)) + D$

  • A: Amplitude (vertical stretch)
  • B: Affects period: Period = $\frac{2\pi}{|B|}$
  • C: Phase shift (horizontal shift)
  • D: Vertical shift

Inverse Trig Functions

Principal Values

  • $\arcsin x$ or $\sin^{-1} x$: Range $[-\frac{\pi}{2}, \frac{\pi}{2}]$
  • $\arccos x$ or $\cos^{-1} x$: Range $[0, \pi]$
  • $\arctan x$ or $\tan^{-1} x$: Range $(-\frac{\pi}{2}, \frac{\pi}{2})$

Trigonometric Identities

Fundamental Identities

Pythagorean Identities

$$\sin^2 \theta + \cos^2 \theta = 1$$

$$\tan^2 \theta + 1 = \sec^2 \theta$$

$$1 + \cot^2 \theta = \csc^2 \theta$$

Quotient Identities

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

$$\cot \theta = \frac{\cos \theta}{\sin \theta}$$

Sum and Difference Formulas

Sine

$$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$$

Cosine

$$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$$

Tangent

$$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$$

Double Angle Formulas

$$\sin(2\theta) = 2\sin\theta\cos\theta$$

$$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$$

$$\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$$

Half Angle Formulas

$$\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$$

$$\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$$

$$\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}$$

Solving Trigonometric Equations

Basic Strategies

Example: Solve $2\sin x - 1 = 0$ for $0 \leq x < 2\pi$

Step 1:

Isolate the trig function: $\sin x = \frac{1}{2}$

Step 2:

Find reference angle: $x = \arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}$

Step 3:

Find all solutions in the interval:

Since sine is positive in Q1 and Q2: $x = \frac{\pi}{6}, \frac{5\pi}{6}$

Using Identities

Example: Solve $\sin 2x = \sin x$ for $0 \leq x < 2\pi$

Step 1:

Use double angle formula: $2\sin x \cos x = \sin x$

Step 2:

Factor: $\sin x(2\cos x - 1) = 0$

Step 3:

Solve each factor:

$\sin x = 0$ gives $x = 0, \pi$

$\cos x = \frac{1}{2}$ gives $x = \frac{\pi}{3}, \frac{5\pi}{3}$

Applications and Word Problems

Navigation and Bearings

Example: Ship Navigation

A ship travels 50 km on a bearing of 030°, then 80 km on a bearing of 120°. Find the distance and bearing from the starting point.

Step 1:

Convert to standard angles: 030° → 60° from east, 120° → 30° from east

Step 2:

Find components:

First leg: $x_1 = 50\cos(60°) = 25$, $y_1 = 50\sin(60°) = 43.3$

Second leg: $x_2 = 80\cos(30°) = 69.3$, $y_2 = 80\sin(30°) = 40$

Step 3:

Total displacement: $x = 94.3$, $y = 83.3$

Distance: $\sqrt{94.3^2 + 83.3^2} = 125.8$ km

Simple Harmonic Motion

Motion described by: $y = A\sin(\omega t + \phi)$ or $y = A\cos(\omega t + \phi)$

  • $A$ = amplitude
  • $\omega$ = angular frequency
  • $\phi$ = phase shift
  • Period = $\frac{2\pi}{\omega}$
  • Frequency = $\frac{\omega}{2\pi}$

Practice Problems

Beginner Level

  1. Find $\sin 150°$ using reference angles
  2. Convert $\frac{3\pi}{4}$ radians to degrees
  3. If $\sin \theta = \frac{3}{5}$ and $\theta$ is in Q1, find $\cos \theta$
  4. Evaluate $\tan 225°$

Intermediate Level

  1. Solve $2\cos x + \sqrt{3} = 0$ for $0 \leq x < 2\pi$
  2. Prove: $\frac{\sin x}{1 + \cos x} = \tan \frac{x}{2}$
  3. Find the period and amplitude of $y = 3\sin(2x - \frac{\pi}{4})$
  4. Simplify: $\sin(x + y)\sin(x - y)$

Advanced Level

  1. Solve $\sin x + \sin 3x = 0$ for all real $x$
  2. Prove: $\cos 3x = 4\cos^3 x - 3\cos x$
  3. Find the maximum value of $f(x) = 3\sin x + 4\cos x$
  4. Solve the system: $\sin x + \sin y = 1$, $\cos x + \cos y = 1$

Interactive Visualizations

Interactive Unit Circle

Explore how sine and cosine relate to the unit circle

Trig Function Transformations

See how parameters affect trig function graphs