Trigonometry

Trigonometry

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The study of relationships between angles and sides of triangles, and the extension to circular functions. Trigonometry is essential for understanding periodic phenomena, waves, and rotations.

Introduction to Trigonometry

Trigonometry literally means "triangle measurement." It began in ancient civilizations for practical purposes like astronomy, navigation, and construction. The Babylonians and Egyptians used trigonometric concepts thousands of years ago, but the modern form developed in ancient Greece.

While trigonometry started with triangles, it extends far beyond. The trigonometric functions (sine, cosine, tangent, etc.) describe relationships on the unit circle and model periodic phenomena. They appear everywhere in nature—from the motion of pendulums to the behavior of light waves.

The connection between trigonometry and complex numbers through Euler's formula reveals that trigonometric functions are essentially exponential functions in disguise, unified through the complex plane.

Complete History

Trigonometry has its origins in ancient astronomy and navigation. The Babylonians and Egyptians used basic trigonometric concepts for astronomical calculations around 2000 BCE. The first systematic study of trigonometry as a mathematical discipline began with the ancient Greeks, particularly Hipparchus of Nicaea (c. 190-120 BCE), who created the first trigonometric table and is often called the "father of trigonometry."

The word "trigonometry" comes from the Greek words "trigonon" (triangle) and "metron" (measure). The ancient Greeks developed trigonometry primarily for astronomical purposes. Claudius Ptolemy (c. 100-170 CE) expanded on Hipparchus's work in his "Almagest," creating more comprehensive trigonometric tables and establishing many fundamental relationships between angles and chords.

During the Islamic Golden Age (8th-14th centuries), trigonometry was significantly advanced. Persian and Arab mathematicians like Al-Battani (858-929) and Nasir al-Din al-Tusi (1201-1274) developed spherical trigonometry and introduced the six trigonometric functions we use today. The work was later transmitted to Europe, where it became essential for navigation during the Age of Exploration.

The modern development of trigonometry was revolutionized by Leonhard Euler (1707-1783), who connected trigonometric functions with complex numbers through Euler's formula: e^(ix) = cos(x) + i·sin(x). This connection unified trigonometry with exponential functions and complex analysis. Today, trigonometry is fundamental to physics (waves, oscillations), engineering (signal processing), computer graphics, and many other fields.

Key Concepts

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The Unit Circle

The unit circle (radius = 1) centered at the origin is the foundation of modern trigonometry. Any point on the circle can be described by an angle θ.

x=cos⁡θ,y=sin⁡θx = \cos\theta, \quad y = \sin\theta

As θ increases, the point traces the circle. This gives trigonometric functions their periodic nature.

Sine and Cosine

For an angle θ on the unit circle, cosine is the x-coordinate and sine is the y-coordinate. In a right triangle, they represent ratios of sides.

cos⁡θ=adjacenthypotenuse,sin⁡θ=oppositehypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

These functions are periodic with period 2π, meaning they repeat every full rotation.

Tangent and Other Functions

Tangent is the ratio of sine to cosine. Other functions include secant, cosecant, and cotangent.

tan⁡θ=sin⁡θcos⁡θ,sec⁡θ=1cos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \sec\theta = \frac{1}{\cos\theta}

These functions have different periodicities and properties, but all derive from the fundamental sine and cosine functions.

Periodicity

Trigonometric functions are periodic, repeating their values at regular intervals. This makes them perfect for modeling cyclic phenomena.

sin⁡(θ+2π)=sin⁡θ,cos⁡(θ+2π)=cos⁡θ\sin(\theta + 2\pi) = \sin\theta, \quad \cos(\theta + 2\pi) = \cos\theta

This periodicity connects to rotations, waves, and oscillations in physics and engineering.

Fundamental Theory

Pythagorean Identity

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

The most fundamental trigonometric identity. It follows directly from the unit circle definition and the Pythagorean theorem. This identity is the foundation for all other trigonometric identities.

Angle Addition Formulas

These formulas express trigonometric functions of sums and differences of angles:

sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta
cos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αsin⁡β\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta

These formulas are essential for simplifying expressions and solving trigonometric equations.

Connection to Complex Numbers

Euler's formula reveals that trigonometric functions are related to complex exponentials:

eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta

This connection allows us to derive trigonometric identities using complex number properties, and vice versa.

Quick Examples

Example 1: Using the Pythagorean Identity

If sin⁡θ=35\sin\theta = \frac{3}{5}, find cos⁡θ\cos\theta:

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
(35)2+cos⁡2θ=1\left(\frac{3}{5}\right)^2 + \cos^2\theta = 1
cos⁡2θ=1−925=1625\cos^2\theta = 1 - \frac{9}{25} = \frac{16}{25}
cos⁡θ=±45\cos\theta = \pm\frac{4}{5}

The sign depends on which quadrant θ is in. The Pythagorean identity relates all trigonometric functions.

Example 2: Double Angle Formulas

Using angle addition, we can derive double angle formulas:

sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta
cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

These formulas are essential for solving trigonometric equations and simplifying expressions.

Example 3: Modeling Periodic Motion

A simple harmonic oscillator (like a spring) has position:

x(t)=Acos⁡(ωt+ϕ)x(t) = A\cos(\omega t + \phi)

Where A is amplitude, ω is angular frequency, and φ is phase. This trigonometric function perfectly models the periodic motion, demonstrating why trigonometry is essential to physics.

Example 4: Solving a Right Triangle

Given a right triangle with one angle of 30° and the opposite side of length 5, find the hypotenuse:

sin⁡(30°)=oppositehypotenuse=12\sin(30°) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2}
5h=12\frac{5}{h} = \frac{1}{2}
h=10h = 10

The hypotenuse is 10 units. This demonstrates how trigonometric ratios connect angles to side lengths in right triangles.

Example 5: Sum-to-Product Identities

Express sin(75°) + sin(15°) as a product:

sin⁡A+sin⁡B=2sin⁡(A+B2)cos⁡(A−B2)\sin A + \sin B = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)
sin⁡(75°)+sin⁡(15°)=2sin⁡(45°)cos⁡(30°)\sin(75°) + \sin(15°) = 2\sin(45°)\cos(30°)
=2⋅22⋅32=62= 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2}

Sum-to-product identities are useful for simplifying trigonometric expressions and solving equations.

Practice Problems

Work through these problems to test your understanding of trigonometric concepts. Solutions are provided below each problem.

Problem 1: Finding Trigonometric Values

If cos(θ) = 4/5 and θ is in the first quadrant, find sin(θ) and tan(θ).

Solution:

Using the Pythagorean identity:

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
sin⁡2θ+(45)2=1\sin^2\theta + \left(\frac{4}{5}\right)^2 = 1
sin⁡θ=1−1625=925=35\sin\theta = \sqrt{1 - \frac{16}{25}} = \sqrt{\frac{9}{25}} = \frac{3}{5}
tan⁡θ=sin⁡θcos⁡θ=3/54/5=34\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{3/5}{4/5} = \frac{3}{4}

Problem 2: Solving a Trigonometric Equation

Solve for x: 2sin²(x) - sin(x) - 1 = 0, where 0 ≤ x < 2π.

Solution:

Let u = sin(x), then: 2u² - u - 1 = 0

u=1±1+84=1±34u = \frac{1 \pm \sqrt{1 + 8}}{4} = \frac{1 \pm 3}{4}
u=1 or u=−12u = 1 \text{ or } u = -\frac{1}{2}

So sin(x) = 1 gives x = π/2, and sin(x) = -1/2 gives x = 7π/6 or 11π/6.

Solutions: x = π/2, 7π/6, 11π/6

Applications

Trigonometry is fundamental to many fields:

Physics

Wave mechanics, oscillations, rotations, and periodic motion

Engineering

Signal processing, control systems, and structural analysis

Navigation

GPS, astronomy, and surveying

Computer Graphics

Rotations, transformations, and 3D rendering

Music

Sound waves, harmonics, and frequency analysis

Architecture

Structural design, angles, and measurements

Fundamental Identities

Explore the key identities that form the foundation of trigonometry.

Pythagorean Identity

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

The Pythagorean identity connects sine and cosine through the unit circle, revealing that for any angle θ, the sum of their squares equals 1. This fundamental relationship underlies all of trigonometry and connects to Euler's formula through the unit circle.

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Resources

External resources for further learning: