Graphs of Trigonometric Functions: Explained with Examples

The graphs of sin, cos and tan are pictures of how these functions behave as the angle grows. Each one repeats forever, producing a regular wave with its own rhythm.

These graphs matter because almost everything that repeats in nature — sound waves, ocean tides, daylight hours, alternating electric current — can be drawn as a sine or cosine curve. If you can read these graphs, you can read the mathematics of waves.

Key Concepts

A periodic function repeats its values at regular intervals. The period is the smallest positive distance after which the graph repeats: 2π for sin and cos, but only π for tan. The amplitude of sin and cos is 1, so the curves swing between −1 and +1. The tan graph has vertical asymptotes — lines the curve approaches but never touches — at x = π/2 + nπ for every integer n. Key points are the peaks, valleys and axis crossings that anchor each sketch.

Formulas

The facts you need for sketching, with angles in radians:

• y = sin x: domain all reals, range −1 to 1, period 2π, odd. Key points on [0, 2π]: (0, 0), (π/2, 1), (π, 0), (3π/2, −1), (2π, 0).
• y = cos x: domain all reals, range −1 to 1, period 2π, even. Key points on [0, 2π]: (0, 1), (π/2, 0), (π, −1), (3π/2, 0), (2π, 1). Since cos x = sin(x + π/2), the cosine graph is the sine graph shifted left by π/2.
• y = tan x: period π, range all reals, odd. Zeros at x = nπ and asymptotes at x = π/2 + nπ. Between two asymptotes it rises smoothly from −∞ to +∞.

Worked Example

Sketch y = sin x for x from 0 to 2π using the key points.

1. Mark the five key x-values: 0, π/2, π, 3π/2, 2π.
2. Find the values: sin 0 = 0, sin(π/2) = 1, sin π = 0, sin(3π/2) = −1, sin 2π = 0.
3. Plot (0, 0), (π/2, 1), (π, 0), (3π/2, −1), (2π, 0).
4. Join them with one smooth wave: up to the peak at (π/2, 1), down through (π, 0) to the valley at (3π/2, −1), and back up to (2π, 0).

That is one complete hill-and-valley wave. Repeating the pattern left and right gives the full sine graph, and shifting it left by π/2 gives the cosine graph.

Common Misconceptions

• "The tangent graph is one continuous curve." Wrong. It has vertical asymptotes at π/2 + nπ, so it is broken into separate branches. Never draw across an asymptote.
• "Sin and tan both have period 2π." Wrong. Tan repeats every π, half as often as sin and cos. Mixing up the periods is a very common exam error.
• "The sine graph starts at a peak." Wrong. Sin 0 = 0, so sine starts at the origin and rises. The curve starting at (0, 1) is cosine.

Real-World Applications

Sine and cosine graphs model every repeating natural process: tides, vibrating guitar strings, the alternating voltage in home electricity, and the daily and yearly swing of temperature. Doctors read heartbeat traces built from these waves. The tangent curve appears in optics and navigation, where angles grow toward 90 degrees and values shoot toward infinity.

In short: sin and cos are smooth waves of period 2π swinging between −1 and 1, pinned by five key points per cycle, while tan repeats every π with asymptotes at π/2 + nπ. Sketch from the key points and never cross an asymptote.