Trigonometric Functions

Explore sine, cosine, and tangent functions interactively. Adjust amplitude, frequency, phase shift, and vertical shift to see how they transform the graphs.

Function

Parameters

Controls

Display

Current Equation

y = sin(x)
Amplitude
1.0
Period
2π/1.0
Phase Shift
0.00
Vertical Shift
0.0

Graph

Unit Circle Connection

The angle rotates as the function progresses through its period

Key Properties

Amplitude
1.0
Max distance from midline
Period
2π/1.0
Length of one complete cycle
Phase Shift
0.00
Horizontal translation
Vertical Shift
0.0
Midline position

Understanding Trigonometric Functions

General Form

y = A·f(Bx - C) + D
  • A (Amplitude): Vertical stretch/compression. |A| is the max distance from midline.
  • B (Frequency): Horizontal compression. Period = 2π/B for sin/cos, π/B for tan.
  • C (Phase Shift): Horizontal shift. Positive C shifts right by C/B units.
  • D (Vertical Shift): Moves the midline up/down by D units.

Function Characteristics

Sine (sin x)

Domain: All reals | Range: [-1, 1]
Period: 2π | Starts at (0, 0)

Cosine (cos x)

Domain: All reals | Range: [-1, 1]
Period: 2π | Starts at (0, 1)

Tangent (tan x)

Domain: x ≠ π/2 + nπ | Range: All reals
Period: π | Has vertical asymptotes

Amplitude & Period

Amplitude measures the height of the wave. For y = A·sin(x), the function oscillates between -A and A.

Period is the horizontal length of one complete cycle. Increasing frequency B makes the function oscillate faster (shorter period).

Phase Shift & Vertical Shift

Phase shift moves the graph left or right. In y = sin(Bx - C), the shift is C/B to the right.

Vertical shift D moves the entire graph up or down, changing the midline from y = 0 to y = D.

Real-World Applications

Sound Waves

Sound is modeled as sine waves. Amplitude represents volume, frequency determines pitch. Musical notes are specific frequencies.

AC Electricity

Alternating current follows a sinusoidal pattern. Voltage oscillates at 60 Hz (US) or 50 Hz (most other countries).

Tides & Seasons

Ocean tides and daylight hours vary sinusoidally. The period for tides is about 12.4 hours, for seasons it's one year.

Spring Motion

A mass on a spring oscillates sinusoidally. The frequency depends on spring constant and mass (Hooke's Law).

Radio Waves

Radio and light are electromagnetic waves modeled by sine functions. Different frequencies create different types of radiation.

Biorhythms

Many biological processes are periodic: heartbeat, breathing, circadian rhythms, and hormone cycles.

Discussion