Mathematics.

trigonometric functions

Trigonometric Functions

Trigonometry45 minDifficulty3 out of 10

You should know: unit circle

Overview

The trigonometric functions — sine, cosine, tangent, and their reciprocals cosecant, secant, cotangent — relate angles to ratios of sides in a right triangle, and more generally to coordinates on the unit circle. They are the essential tool for describing anything periodic: waves, oscillations, rotations, and circular motion.

Intuition

In a right triangle, each trig function is a ratio of two sides relative to an angle θ: sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, tangent is opposite/adjacent (SOH-CAH-TOA). Extending this to the unit circle lets these ratios be defined for ANY angle, not just those inside a triangle — including negative angles and angles beyond 90°.

Interactive Graph

Explore sin, cos, tan and their periods

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Formal Definition

Definition

For a right triangle with angle θ, or equivalently a point (cos θ, sin θ) on the unit circle:

sin⁡θ=oppositehypotenuse,cos⁡θ=adjacenthypotenuse,tan⁡θ=sin⁡θcos⁡θ=oppositeadjacent\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{\text{opposite}}{\text{adjacent}}
Right-triangle definitions
csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}

The three reciprocal functions

Notation

NotationMeaning
sin⁡,cos⁡,tan⁡\sin, \cos, \tanThe three primary trigonometric functions
csc⁡,sec⁡,cot⁡\csc, \sec, \cotTheir reciprocals

Derivation

The graph of sin(θ) traces the y-coordinate of a point moving around the unit circle at constant angular speed, producing a smooth oscillation between -1 and 1 with period 2π:

θ:0→π2→π→3π2→2π\theta: 0 \to \frac{\pi}{2} \to \pi \to \frac{3\pi}{2} \to 2\pi
sin⁡θ:0→1→0→−1→0\sin\theta: 0 \to 1 \to 0 \to -1 \to 0

One full period

Properties

Pythagorean identity

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

Periodicity

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

Range

−1≤sin⁡θ≤1,−1≤cos⁡θ≤1-1 \leq \sin\theta \leq 1, \quad -1 \leq \cos\theta \leq 1

Domain restriction of tan

tan⁡θ undefined at θ=π2+kπ\tan\theta \text{ undefined at } \theta = \frac{\pi}{2} + k\pi

Applications

Simple harmonic motion, wave equations, and AC electrical circuits are all modeled with sine and cosine.

Animation

Animates a point traveling around the unit circle while simultaneously tracing out the sine wave as a function of the swept angle, showing the direct link between circular motion and the graph.

Worked Examples

  1. These are standard angle values from the 30-60-90 triangle.

    sin⁡(30°)=12,cos⁡(30°)=32\sin(30°) = \frac{1}{2}, \quad \cos(30°) = \frac{\sqrt{3}}{2}

Answer: sin(30°) = 1/2, cos(30°) = √3/2

Practice Problems

Difficulty 3/10

Find tan(45°).

Difficulty 4/10

A loading ramp is 12 m long and rises to a dock 3 m high. What angle does the ramp make with the ground?

Difficulty 5/10

From 50 m away, the angle of elevation to the top of a building is 35°. How tall is the building (ignoring eye height)?

Common Mistakes

Common Mistake

Assuming tan(θ) is defined for every angle.

tan(θ) = sin(θ)/cos(θ) is undefined wherever cos(θ) = 0, i.e. at θ = π/2 + kπ.

Common Mistake

Mixing up which side is 'opposite' vs 'adjacent'.

Opposite and adjacent are always relative to the specific angle θ being used — they swap if you pick the other acute angle in the triangle.

Quiz

What is the period of sin(θ) and cos(θ)?

Flashcards

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Historical Background

Trigonometry originated in the astronomy of the ancient world. Hipparchus built the first known trigonometric table around 140 BCE. Indian mathematicians, notably Aryabhata, introduced the sine function directly (as opposed to the Greek chord function) around 500 CE, and this sine-based approach passed through Islamic Golden Age mathematics into Europe, where it was standardized into the six modern functions by the Renaissance.

  1. c. 140 BCE

    Hipparchus compiles a table of chords

    Hipparchus

  2. c. 500 CE

    Aryabhata defines the sine (jya) function directly

    Aryabhata

  3. 16th century

    European mathematicians standardize the six trigonometric functions

Summary

  • sin, cos, tan relate an angle to ratios of triangle sides, or coordinates on the unit circle.
  • csc, sec, cot are their reciprocals.
  • sin²θ + cos²θ = 1 always holds (Pythagorean identity).
  • sin and cos have period 2π and range [-1,1]; tan has period π and is undefined at odd multiples of π/2.

References

  1. WebsiteWikipedia — Trigonometric functions