Graphs of Sine, Cosine and Tangent Functions
The graphs of y = sin x and y = cos x are smooth waves between −1 and 1 with period 2π, and the graph of y = tan x repeats every π with vertical asymptotes where cos x = 0. In brief: sine starts at (0, 0), reaches its maximum 1 at x = π/2 and returns to 0 at π; cosine starts at (0, 1). The cosine graph is the sine graph shifted left by π/2. Tangent passes through the origin and increases between asymptotes at x = ±π/2.
These three graphs are the basic models of periodic behaviour: tides, sound waves, seasons and alternating current. Knowing their key points lets you sketch them in seconds and read their properties. This guide describes each graph, its domain, range, period, symmetry, zeros and extreme points, and gives value tables with a figure of sine and cosine.
What does the graph of y = sin x look like?
The graph is obtained by plotting the y-coordinate of a point moving around the unit circle against the angle x (in radians). Key features:
- Domain: all real numbers. Range: [−1, 1].
- Period 2π: sin(x + 2π) = sin x.
- Zeros at x = kπ (k an integer). Maximum 1 at x = π/2 + 2kπ and minimum −1 at x = 3π/2 + 2kπ.
- Odd function: sin(−x) = −sin x, so the graph is symmetric about the origin.
- Increasing on [−π/2, π/2] and decreasing on [π/2, 3π/2].
What does the graph of y = cos x look like?
- Domain: all real numbers. Range: [−1, 1]. Period 2π.
- The graph starts at its maximum (0, 1). Zeros at x = π/2 + kπ. Maximum 1 at x = 2kπ and minimum −1 at x = π + 2kπ.
- Even function: cos(−x) = cos x, so the graph is symmetric about the y-axis.
- cos x = sin(x + π/2): the cosine graph is the sine graph shifted π/2 to the left.
What does the graph of y = tan x look like?
- Domain: all real x except x = π/2 + kπ, where cos x = 0. Range: all real numbers.
- Period π (not 2π): tan(x + π) = tan x.
- Zeros at x = kπ. Vertical asymptotes at x = π/2 + kπ.
- Odd function and increasing on each interval between asymptotes.
- Key values: tan(π/4) = 1 and tan(−π/4) = −1.
The graph near zero looks like the line y = x: tan 0.1 ≈ 0.1003. It rises steeply near the asymptotes: tan 1.5 ≈ 14.1.
Value table for one period
| x | sin x | cos x | tan x |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| π/6 | 0.5 | 0.866 | 0.577 |
| π/4 | 0.707 | 0.707 | 1 |
| π/3 | 0.866 | 0.5 | 1.732 |
| π/2 | 1 | 0 | undefined |
| 2π/3 | 0.866 | −0.5 | −1.732 |
| 3π/4 | 0.707 | −0.707 | −1 |
| 5π/6 | 0.5 | −0.866 | −0.577 |
| π | 0 | −1 | 0 |
The remaining half of the period for sine and cosine follows from the symmetries: sin(x + π) = −sin x and cos(x + π) = −cos x.
Properties compared
| Property | y = sin x | y = cos x | y = tan x |
|---|---|---|---|
| Domain | all reals | all reals | x ≠ π/2 + kπ |
| Range | [−1, 1] | [−1, 1] | all reals |
| Period | 2π | 2π | π |
| Parity | odd | even | odd |
| Zeros | kπ | π/2 + kπ | kπ |
| y-intercept | 0 | 1 | 0 |
| Asymptotes | none | none | x = π/2 + kπ |
How do you sketch the graphs quickly?
- Mark one period on the x-axis and divide it into four equal parts: for sine and cosine, these are the quarter points 0, π/2, π, 3π/2, 2π.
- Sine takes the values 0, 1, 0, −1, 0 at these points. Cosine takes the values 1, 0, −1, 0, 1.
- Plot the five points and connect them with a smooth wave, rounding the peaks.
- Repeat to the left and right. For tangent, draw the asymptotes first, then the curve through (0, 0) with the points (±π/4, ±1).
The transformed versions, with different amplitude, period and shifts.
Where do these graphs come from?
The sine curve appeared in the study of vibrating strings in the eighteenth century, in the work of Brook Taylor, Daniel Bernoulli and Leonhard Euler on the shape of a string. The idea that the curve is the graph of a function came with the general function concept. In the nineteenth century Joseph Fourier showed that more general periodic functions can be written as sums of sines and cosines.
Where are they used today?
The sine wave describes pure musical tones, with the frequency as the pitch. Alternating current in households follows a sinusoid with 50 or 60 cycles per second. Tides, daylight hours and the seasonal changes in temperature are approximately sinusoidal. The slope of a road is a tangent, and the tangent graph shows why slopes become very steep near the vertical.
Which mistakes are common?
Drawing the tangent graph through the asymptotes
The curve approaches the asymptotes but never crosses them. Draw the branches separately.
Mixing up sine and cosine at x = 0
Sine starts at 0, cosine starts at 1.
Using degrees on the x-axis by mistake
With x in radians, one period is about 6.28 units long, not 360. Label the axis with π.
Forgetting that tangent has period π
Tangent repeats twice as fast as sine and cosine.