1. Foundations of Trigonometry

What is Trigonometry?

Trigonometry (from Greek trigōnon "triangle" and metron "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. It has vast applications in science, engineering, navigation, and many other fields.

Historical Origins

Ancient India

Indian mathematicians developed trigonometry as an astronomical tool around 5th century CE. Aryabhata created sine tables.

Ancient Greece

Hipparchus (190-120 BCE) compiled the first trigonometric table and is called the "father of trigonometry".

Islamic Golden Age

Al-Khwarizmi and Al-Kashi computed detailed trigonometric tables and developed spherical trigonometry.

Angle Measurement

Degree: A full circle = 360° (°)

Radian: A full circle = 2π radians (rad)

$$1^\circ = \frac{\pi}{180} \text{ rad} \quad \text{or} \quad 1 \text{ rad} = \frac{180}{\pi}^\circ$$

Conversion Formulas

Degrees to Radians
$$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$$

Example: Convert 45° to radians:

$$45 \times \frac{\pi}{180} = \frac{\pi}{4} \text{ rad}$$

Radians to Degrees
$$\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$$

Example: Convert π/3 rad to degrees:

$$\frac{\pi}{3} \times \frac{180}{\pi} = 60^\circ$$

The Unit Circle

The unit circle is a circle with radius 1 centered at the origin (0, 0). It's fundamental to understanding trigonometric functions for all angles.

Key Points on Unit Circle:

  • (1, 0) → 0° or 0 rad
  • (0, 1) → 90° or π/2 rad
  • (-1, 0) → 180° or π rad
  • (0, -1) → 270° or 3π/2 rad

    • 2. Basic Trigonometric Ratios

    In a right triangle with angle θ, the three basic trigonometric ratios are defined as follows:

    Sine (sin θ)

    $$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{AC}$$

    Represents the ratio of the side opposite to angle θ to the hypotenuse.

    Cosine (cos θ)

    $$\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{BC}{AC}$$

    Represents the ratio of the side adjacent to angle θ to the hypotenuse.

    Tangent (tan θ)

    $$\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} = \frac{\sin\theta}{\cos\theta}$$

    Represents the ratio of the opposite side to the adjacent side.

    Reciprocal Functions

    Cosecant

    $$\csc\theta = \frac{1}{\sin\theta} = \frac{\text{Hypotenuse}}{\text{Opposite}}$$

    Secant

    $$\sec\theta = \frac{1}{\cos\theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$$

    Cotangent

    $$\cot\theta = \frac{1}{\tan\theta} = \frac{\text{Adjacent}}{\text{Opposite}}$$

    3. Standard Table of Trigonometric Values

    Angle sin θ cos θ tan θ
    0° (0)010
    30° (π/6)1/2√3/21/√3
    45° (π/4)√2/2√2/21
    60° (π/3)√3/21/2√3
    90° (π/2)10∞
    120° (2π/3)√3/2-1/2-√3
    135° (3π/4)√2/2-√2/2-1
    150° (5π/6)1/2-√3/2-1/√3
    180° (π)0-10
    270° (3π/2)-10∞
    360° (2π)010

    Quick Reference: Special Angles

    sin 0° = 0

    sin 30° = 1/2

    sin 45° = √2/2

    sin 60° = √3/2

    sin 90° = 1

    cos 0° = 1

    cos 30° = √3/2

    cos 45° = √2/2

    cos 60° = 1/2

    cos 90° = 0

    tan 0° = 0

    tan 30° = 1/√3

    tan 45° = 1

    tan 60° = √3

    tan 90° = ∞

    4. Trigonometric Identities

    Fundamental Identities

    Pythagorean Identity

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

    Tangent Identity

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

    Cotangent Identity

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

    Reciprocal Identities

    $$\sin\theta = \frac{1}{\csc\theta}, \quad \cos\theta = \frac{1}{\sec\theta}, \quad \tan\theta = \frac{1}{\cot\theta}$$
    $$\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}$$

    Quotient Identities

    $$\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}$$

    Compound Angle Formulas

    Sine of Sum/Difference
    $$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$$

    Example: sin(60° + 30°) = sin 60° cos 30° + cos 60° sin 30°

    = (√3/2)(√3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1 ✓

    Cosine of Sum/Difference
    $$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$$
    Tangent of Sum/Difference
    $$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$$

    Double Angle Formulas

    sin 2θ

    $$\sin 2\theta = 2\sin\theta\cos\theta$$

    cos 2θ

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

    tan 2θ

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

    Half Angle Formulas

    $$\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}$$
    $$\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}$$
    $$\tan\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} = \frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}$$

    5. Graphs of Trigonometric Functions

    Interactive Function Graph

    Function Properties

    sin(x)

    Domain: ℝ

    Range: [-1, 1]

    Period: 2π

    Type: Odd

    cos(x)

    Domain: ℝ

    Range: [-1, 1]

    Period: 2π

    Type: Even

    tan(x)

    Domain: ℝ - (π/2 + kπ)

    Range: ℝ

    Period: π

    Type: Odd

    6. Inverse Trigonometric Functions

    Inverse trigonometric functions reverse the operation of the trigonometric functions. They return the angle whose trig value is given.

    arcsin(x) = sin⁻¹x

    $$y = \sin^{-1}x \iff \sin y = x$$

    Domain: [-1, 1]

    Range: [-π/2, π/2]

    arccos(x) = cos⁻¹x

    $$y = \cos^{-1}x \iff \cos y = x$$

    Domain: [-1, 1]

    Range: [0, π]

    arctan(x) = tan⁻¹x

    $$y = \tan^{-1}x \iff \tan y = x$$

    Domain: ℝ

    Range: (-π/2, π/2)

    Key Properties

    $$\sin(\sin^{-1}x) = x, \quad \cos(\cos^{-1}x) = x, \quad \tan(\tan^{-1}x) = x$$
    $$\sin^{-1}(\sin\theta) = \theta \text{ (if } \theta \in [-\pi/2, \pi/2])$$
    $$\cos^{-1}x + \sin^{-1}x = \frac{\pi}{2}$$

    7. Trigonometric Equations

    Trigonometric equations involve trigonometric functions of an unknown angle. We solve for the angle(s) that satisfy the equation.

    Solving Basic Equations

    Example 1: sin x = 1/2

    Step 1: Find principal value

    sin⁻¹(1/2) = 30° or π

    Example 2: cos x = -1/2

    Step 1: Principal value

    cos⁻¹(-1/2) = 120° or 2π/3

    Step 2: General solution

    x = 120° + 360°k or x = 240° + 360°k

    or x = 2π/3 + 2πk or x = 4π/3 + 2πk

    General Solutions

    sin x = sin α → x = nπ + (-1)ⁿα

    cos x = cos α → x = 2nπ ± α

    tan x = tan α → x = nπ + α

    8. Applications of Trigonometry

    📐

    Height & Distance

    Calculate heights of buildings, mountains, and trees using angles of elevation and depression.

    $$\tan\theta = \frac{\text{height}}{\text{distance}}$$
    🧭

    Navigation

    GPS, compass bearings, and celestial navigation use trigonometry for positioning.

    🌊

    Waves & Oscillations

    Sound waves, light waves, and tides are modeled using sinusoidal functions.

    $$y = A\sin(\omega t + \phi)$$
    🏗️

    Engineering

    Structural analysis, electrical circuits, and mechanical systems rely on trig calculations.

    🔭

    Astronomy

    Calculating distances to stars, planetary positions, and orbital mechanics.

    🎮

    Computer Graphics

    3D rendering, game development, and animations use rotation matrices.

    Practice Problem

    Problem: From a point 50 meters from the base of a tree, the angle of elevation to the top is 30°. Find the height of the tree.

    Solution:

    tan(30°) = height / 50

    1/√3 = h / 50

    h = 50/√3 ≈ 28.87 meters

    9. Advanced Topics

    Euler's Formula

    $$e^{i\theta} = \cos\theta + i\sin\theta$$

    This beautiful formula connects complex exponentials with trigonometric functions.

    Complex Numbers Connection

    Using Euler's formula, we can express any complex number in trigonometric form:

    $$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$$

    where r = |z| (modulus) and θ = arg(z) (argument)

    Hyperbolic Functions

    Hyperbolic Sine

    $$\sinh x = \frac{e^x - e^{-x}}{2}$$

    Hyperbolic Cosine

    $$\cosh x = \frac{e^x + e^{-x}}{2}$$

    Hyperbolic Tangent

    $$\tanh x = \frac{\sinh x}{\cosh x}$$

    Trigonometric Substitution

    Used to integrate expressions containing √(a² - x²), √(a² + x²), or √(x² - a²)

    For √(a² - x²): Let x = a sinθ

    For √(a² + x²): Let x = a tanθ

    For √(x² - a²): Let x = a secθ

    Fourier Series Introduction

    Any periodic function can be expressed as a sum of sine and cosine waves:

    $$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} [a_n\cos(nx) + b_n\sin(nx)]$$

    10. Practice & Testing