Introduction To Trigonometry

Introduction to Trigonometry

Definition

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right-angled triangles. The word “trigonometry” comes from the Greek words “trigonon” (triangle) and “metron” (measure), meaning “measurement of triangles.”

Key Concept:
Trigonometry focuses on understanding how angles and their corresponding ratios (such as sine, cosine, and tangent) describe geometric properties and relationships.
Significance in Mathematics:
Trigonometry is fundamental in solving problems involving triangles and periodic phenomena. It extends beyond pure mathematics to practical applications in science, engineering, and technology.

Historical Background

Origins:
Trigonometry has its roots in ancient civilizations such as Babylon and Egypt, where it was used for astronomy and building structures like pyramids.
Development:
- The Greeks, particularly Hipparchus (known as the “Father of Trigonometry”), formalized trigonometric concepts.
- Indian mathematicians like Aryabhata and Brahmagupta contributed significantly to the development of sine and cosine functions.
- Arabic scholars preserved and expanded on Greek and Indian trigonometry, introducing terms like “sine” and “tangent.”
- During the European Renaissance, mathematicians like Johannes Kepler and Leonhard Euler advanced trigonometry for use in physics and astronomy.

Applications of Trigonometry

Geometry:
- Solving for unknown sides or angles in triangles.
- Analyzing shapes like circles, polygons, and ellipses.
Physics:
- Describing wave motion and oscillations.
- Calculating forces, velocities, and displacements in mechanics.
Engineering:
- Designing and analyzing structures such as bridges and buildings.
- Creating accurate blueprints and CAD (computer-aided design) models.
Astronomy:
- Measuring distances between celestial bodies.
- Calculating the orbits of planets and satellites.
Navigation:
- Determining positions and plotting courses using triangulation.
- Calculating angles of elevation or depression for surveying.

Real-Life Examples

Measuring Heights of Buildings or Trees:
Using the angle of elevation and a known distance, trigonometry helps calculate the height of tall objects.
Surveying and Mapping:
Trigonometry is used to determine distances and angles between land features.
Flight and Maritime Navigation:
Pilots and sailors use trigonometry to calculate direction and distance.
Wave Analysis:
Trigonometric functions model sound waves, light waves, and alternating current in electrical engineering.

Trigonometric Ratios

Definition

Trigonometric ratios are mathematical relationships between the lengths of the sides of a right triangle. These ratios are defined relative to one of the acute angles, $\boldsymbol{\theta}$ , in the triangle. Trigonometric ratios form the foundation of trigonometry and are crucial for solving problems involving triangles, angles, and distances.

The Six Trigonometric Ratios

In a right triangle, let:

The angle be $\boldsymbol{\theta}$ .
The sides be labeled as:
- Opposite Side: The side opposite to the angle $\boldsymbol{\theta}$ .
- Adjacent Side: The side next to the angle $\boldsymbol{\theta}$ (excluding the hypotenuse).
- Hypotenuse: The longest side of the triangle, opposite the right angle.

The six trigonometric ratios are defined as follows:

Sine ( $\boldsymbol{\sin \theta}$ ):
$\displaystyle\boldsymbol{\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}}$
Cosine ( $\boldsymbol{\cos \theta}$ ):
$\displaystyle\boldsymbol{\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}}$
Tangent ( $\boldsymbol{\tan \theta}$ ):
$\displaystyle\boldsymbol{\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}}$
Cosecant ( $\boldsymbol{\csc \theta}$ ):
$\displaystyle\boldsymbol{\csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite Side}}}$
Secant ( $\boldsymbol{\sec \theta}$ ):
$\displaystyle\boldsymbol{\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent Side}}}$
Cotangent ( $\boldsymbol{\cot \theta}$ ):
$\displaystyle\boldsymbol{\cot \theta = \frac{\text{Adjacent Side}}{\text{Opposite Side}}}$

Right Triangle Reference

To clearly understand the trigonometric ratios, consider a right triangle $\boldsymbol{\triangle ABC}$ :

$\boldsymbol{\angle B}$ is the right angle.
$\boldsymbol{\theta = \angle A}$ .

The sides are labeled as:

Opposite Side: $\boldsymbol{\text{BC}}$ (opposite to $\boldsymbol{\angle A}$ ).
Adjacent Side: $\boldsymbol{\text{AB}}$ (next to $\boldsymbol{\angle A}$ , excluding the hypotenuse).
Hypotenuse: $\boldsymbol{\text{AC}}$ (longest side, opposite the right angle).

Key Observations

Trigonometric ratios depend on the angle $\boldsymbol{\theta}$ and not on the size of the triangle.
The reciprocal relationships between the ratios are:
- $\displaystyle\boldsymbol{\csc \theta = \frac{1}{\sin \theta}}$
- $\displaystyle\boldsymbol{\sec \theta = \frac{1}{\cos \theta}}$
- $\displaystyle\boldsymbol{\cot \theta = \frac{1}{\tan \theta}}$

Trigonometric Ratios of Standard Angles

Angles

The standard angles in trigonometry are:
$\boldsymbol{0^\circ}$ , $\boldsymbol{30^\circ}$ , $\boldsymbol{45^\circ}$ , $\boldsymbol{60^\circ}$ , and $\boldsymbol{90^\circ}$ .
These angles are frequently used in trigonometric calculations and have specific trigonometric ratio values.

Table of Ratios

The values of the six trigonometric ratios for the standard angles are:

$\boldsymbol{\theta}$ (Angle)	$\boldsymbol{\sin \theta}$	$\boldsymbol{\cos \theta}$	$\boldsymbol{\tan \theta}$	$\boldsymbol{\csc \theta}$	$\boldsymbol{\sec \theta}$	$\boldsymbol{\cot \theta}$
$\boldsymbol{0^\circ}$	$\boldsymbol{0}$	$\boldsymbol{1}$	$\boldsymbol{0}$	$\boldsymbol{\text{Undefined}}$	$\boldsymbol{1}$	$\boldsymbol{\text{Undefined}}$
$\boldsymbol{30^\circ}$	$\displaystyle\boldsymbol{\frac{1}{2}}$	$\displaystyle\boldsymbol{\frac{\sqrt{3}}{2}}$	$\displaystyle\boldsymbol{\frac{1}{\sqrt{3}}}$	$\boldsymbol{2}$	$\displaystyle\boldsymbol{\frac{2}{\sqrt{3}}}$	$\boldsymbol{\sqrt{3}}$
$\boldsymbol{45^\circ}$	$\displaystyle\boldsymbol{\frac{1}{\sqrt{2}}}$	$\displaystyle\boldsymbol{\frac{1}{\sqrt{2}}}$	$\boldsymbol{1}$	$\boldsymbol{\sqrt{2}}$	$\boldsymbol{\sqrt{2}}$	$\boldsymbol{1}$
$\boldsymbol{60^\circ}$	$\displaystyle\boldsymbol{\frac{\sqrt{3}}{2}}$	$\displaystyle\boldsymbol{\frac{1}{2}}$	$\boldsymbol{\sqrt{3}}$	$\displaystyle\boldsymbol{\frac{2}{\sqrt{3}}}$	$\boldsymbol{2}$	$\displaystyle\boldsymbol{\frac{1}{\sqrt{3}}}$
$\boldsymbol{90^\circ}$	$\boldsymbol{1}$	$\boldsymbol{0}$	$\boldsymbol{\text{Undefined}}$	$\boldsymbol{1}$	$\boldsymbol{\text{Undefined}}$	$\boldsymbol{0}$

Memorization Tips

For $\boldsymbol{\sin \theta}$ and $\boldsymbol{\cos \theta}$ :
- Values of $\boldsymbol{\sin \theta}$ follow the sequence:
  $\displaystyle\boldsymbol{\sin 0^\circ = 0, \sin 30^\circ = \frac{1}{2}, \sin 45^\circ = \frac{1}{\sqrt{2}}, \sin 60^\circ = \frac{\sqrt{3}}{2}, \sin 90^\circ = 1}$
- Values of $\boldsymbol{\cos \theta}$ are the reverse of $\boldsymbol{\sin \theta}$ :
  $\displaystyle\boldsymbol{\cos 0^\circ = 1, \cos 30^\circ = \frac{\sqrt{3}}{2}, \cos 45^\circ = \frac{1}{\sqrt{2}}, \cos 60^\circ = \frac{1}{2}, \cos 90^\circ = 0}$
For $\boldsymbol{\tan \theta}$ :
- $\displaystyle\boldsymbol{\tan \theta = \frac{\sin \theta}{\cos \theta}}$ . Memorizing $\boldsymbol{\sin \theta}$ and $\boldsymbol{\cos \theta}$ makes calculating $\boldsymbol{\tan \theta}$ easier.
Reciprocal Ratios:
- $\displaystyle\boldsymbol{\csc \theta = \frac{1}{\sin \theta}}$ , $\boldsymbol{\sec \theta = \frac{1}{\cos \theta}}$ , $\boldsymbol{\cot \theta = \frac{1}{\tan \theta}}$ .
Mnemonic Device:
- For $\boldsymbol{\sin \theta}$ :
  "Some People Have Curious Senses" ( $\displaystyle\boldsymbol{0, \frac{1}{2}, \frac{1}{\sqrt{2}}, \frac{\sqrt{3}}{2}, 1}$ ).
- For $\boldsymbol{\cos \theta}$ :
  "Cats Sleep Perfectly Halved Opposite" ( $\displaystyle\boldsymbol{1, \frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}, \frac{1}{2}, 0}$ ).

Trigonometric Identities

Definition

Trigonometric identities are mathematical equations that are universally true for all values of $\boldsymbol{\theta}$ , where the expressions involved are defined. These identities establish relationships between trigonometric functions and are fundamental tools for simplifying expressions and solving equations in trigonometry.

Key Identities

Pythagorean Identities: These are derived from the Pythagorean theorem applied to a right triangle:
- $\boldsymbol{\sin^2 \theta + \cos^2 \theta = 1}$
- $\boldsymbol{1 + \tan^2 \theta = \sec^2 \theta}$
- $\boldsymbol{1 + \cot^2 \theta = \csc^2 \theta}$

Applications of Trigonometric Identities

Simplifying Trigonometric Expressions:
- Example: Simplify $\boldsymbol{\sec^2 \theta - \tan^2 \theta}$ .
  Using the identity $\boldsymbol{\sec^2 \theta - \tan^2 \theta = 1}$ , the simplified expression is $\boldsymbol{1}$ .
Solving Trigonometric Equations:
- Example: Solve $\boldsymbol{\sin^2 \theta = 1 - \cos^2 \theta}$ .
  Using the identity $\boldsymbol{\sin^2 \theta + \cos^2 \theta = 1}$ , substitute and solve for $\boldsymbol{\theta}$ .
Verifying Identities:
- Example: Prove $\displaystyle\boldsymbol{\frac{\sin^2 \theta}{1 - \cos^2 \theta} = 1}$ .
  Using $\boldsymbol{1 - \cos^2 \theta = \sin^2 \theta}$ , the left-hand side simplifies to $\boldsymbol{1}$ .
Applications in Geometry and Physics:
- Simplify wave equations, analyze circular motion, and solve problems in mechanics and optics.

Tips for Using Identities

Memorize the key identities as they form the foundation for most trigonometric problems.
Practice rewriting expressions in terms of $\boldsymbol{\sin \theta}$ and $\boldsymbol{\cos \theta}$ to simplify.
Use the identities to convert between functions like $\boldsymbol{\tan \theta}$ , $\boldsymbol{\csc \theta}$ , $\boldsymbol{\sec \theta}$ , and $\boldsymbol{\cot \theta}$ .

Applications of Trigonometry

1. Height and Distance Problems

Trigonometry is extensively used to solve problems involving heights and distances. The primary trigonometric ratios $\boldsymbol{\sin \theta}$ , $\boldsymbol{\cos \theta}$ , and $\boldsymbol{\tan \theta}$ are used to calculate the height of an object or the distance between two points.

Key Concepts:

Angle of Elevation:
The angle between the horizontal line and the line of sight when looking upward at an object.
- Example: Measuring the height of a tower using $\displaystyle\boldsymbol{\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}}$ .
Angle of Depression:
The angle between the horizontal line and the line of sight when looking downward at an object.
- Example: Calculating the distance of a ship from a lighthouse using $\boldsymbol{\tan \theta}$ .

Example: Calculating Height

A person standing 50 meters away from the base of a tower observes the top of the tower at an angle of elevation of $\boldsymbol{30^\circ}$ . Find the height of the tower.
Using $\displaystyle\boldsymbol{\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}}$ :
$\displaystyle\boldsymbol{\tan 30^\circ = \frac{\text{Height}}{50}}$
$\displaystyle\boldsymbol{\frac{1}{\sqrt{3}} = \frac{\text{Height}}{50}}$
$\displaystyle\boldsymbol{\text{Height} = \frac{50}{\sqrt{3}} \approx 28.87 , \text{meters}}$ .

2. Real-Life Applications

Trigonometry is a critical tool in various real-world scenarios. Here are some of the key applications:

Construction and Architecture:

Structural Design:
Engineers use trigonometry to calculate slopes, angles, and heights in designing buildings, bridges, and roofs.
Blueprint Measurements:
Trigonometric functions help ensure accurate dimensions in construction plans.

Navigation:

Positioning and Triangulation:
Pilots and sailors use trigonometric principles to determine their location and plot courses.
- Example: Calculating the distance of a ship from the shore by measuring the angle of elevation to a known height, such as a lighthouse.

Astronomy:

Measuring Distances:
Astronomers use trigonometric parallax to measure the distance to stars and other celestial bodies.
Tracking Orbits:
Trigonometry helps analyze the motion of planets, moons, and satellites.

Physics:

Wave Motion:
Trigonometric functions model oscillatory motion, such as sound waves, light waves, and alternating current.
Projectile Motion:
Trigonometry is used to analyze angles and distances in the trajectory of projectiles.

Example: Calculating Angle of Depression

A drone is flying at a height of 100 meters and observes a car on the ground at an angle of depression of $\boldsymbol{45^\circ}$ . Find the horizontal distance between the drone and the car.
Using $\displaystyle\boldsymbol{\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}}$ :
$\displaystyle\boldsymbol{\tan 45^\circ = \frac{100}{\text{Distance}}}$
$\displaystyle\boldsymbol{1 = \frac{100}{\text{Distance}}}$
$\boldsymbol{\text{Distance} = 100 , \text{meters}}$ .

Introduction To Trigonometry