Trigonometric graphs

The sine and the cosine assign a number to every angle, not only to the angles inside a triangle. That makes them functions — and every function can be drawn.

The circle unrolled into a wave

Picture a point travelling around the unit circle. Put the angle on the horizontal axis and the height of the point — its $y$-coordinate — on the vertical one. What you get is the graph of $y = \sin x$.

The height climbs from $0$ (angle $0$) to $1$ (angle $\frac{\pi}{2}$), returns to $0$ (angle $\pi$), falls to $-1$ (angle $\frac{3\pi}{2}$) and is back at zero after a full turn. Then everything repeats forever — because the point on the circle also starts another lap.

The argument axis is scaled in radians — hence the ticks at $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$ instead of $1, 2, 3$. If you would rather think in degrees, convert them with our angle converter: $\pi$ is $180^\circ$, and $2\pi$ is a full turn.

Period, amplitude, range

Three properties are visible on the graph at once.

The period is the length of one complete run, after which the graph repeats. For the sine and the cosine it is $2\pi$, the length of one lap around the circle:

$$\sin(x + 2\pi) = \sin x, \qquad \cos(x + 2\pi) = \cos x$$

The amplitude is half the distance between the highest and the lowest value — for both functions it is $1$, the radius of the unit circle.

The range is the interval from $-1$ to $1$. The graph never leaves the horizontal lines $y = 1$ and $y = -1$, because a coordinate of a point on a circle of radius $1$ cannot be larger.

The cosine is a shifted sine

The two graphs have an identical shape. They differ only in position: the cosine starts at $1$, the sine at zero.

$$\cos x = \sin\left(x + \frac{\pi}{2}\right)$$

The explanation is on the unit circle: the cosine measures the point's motion horizontally, the sine vertically. It is the same rotation seen from two perpendicular directions, so one run leads the other by $90^\circ$. In physics one says the waves are out of phase.

The zeros interleave: the sine vanishes at $x = k\pi$, the cosine at $x = \frac{\pi}{2} + k\pi$ — exactly where the other one peaks.

The graph of the tangent

The tangent is a quotient, so its graph looks nothing like a wave:

$$\tan x = \frac{\sin x}{\cos x}$$

Three differences from the sine wave:

• The period is $\pi$, not $2\pi$ — the graph repeats twice as often: $\tan(x + \pi) = \tan x$.
• The range is all real numbers — the tangent is bounded neither above nor below.
• The graph has asymptotes — vertical lines the branches approach without ever touching.

The asymptotes stand where the cosine vanishes:

$$x = \frac{\pi}{2} + k\pi \qquad \text{that is} \qquad 90^\circ, \; 270^\circ, \; -90^\circ, \; \ldots$$

Division by zero is impossible, so $\tan 90^\circ$ does not exist — it is not "a very large number" but the absence of a value. The closer you come to $90^\circ$, the larger the tangent grows: $\tan 89^\circ \approx 57$, $\tan 89.9^\circ \approx 573$. They rise without bound, yet $90^\circ$ itself is never reached.

Where these graphs show up

The sine wave is not a school invention — it is the shape almost every repeating phenomenon takes:

• mains voltage — alternating current at $50\,\text{Hz}$ varies exactly sinusoidally;
• sound — a pure tone is a sine wave; we hear its frequency as pitch and its amplitude as loudness;
• tides, pendulums, a vibrating spring — all described by the same function;
• a radio signal — the carrier wave is a sine wave with information laid on top of it.

Fourier analysis goes one step further: any periodic signal — even a square digital one — can be written as a sum of sines of different frequencies. Audio and image compression are built on it.

How changing the formula to $y = A \sin(Bx)$ stretches and squeezes these graphs belongs to the transformations of functions — it returns in the branch on functions and analysis.

Exercises

The questions check what you read off the graph itself: the value of a function at a special angle — including a negative angle and one past a full turn, which is exactly what periodicity means. Type the exact value, e.g. √2/2. Converting degrees to radians is what you practised on the unit circle.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

cos 90° =ResultCheck

Common mistakes

• Quoting the period of the tangent as $2\pi$ — the tangent repeats every $\pi$.
• Treating an asymptote as a zero — at $x = \frac{\pi}{2}$ the tangent has no value, rather than the value zero.
• Writing $\tan 90^\circ = \infty$ — the function has no value there; infinity is not a number.
• An axis scaled in degrees under a formula in radians — if the axis shows $\pi$, the argument is a radian.
• Swapping the sine and the cosine — the graph that starts at $1$ for $x = 0$ is the cosine.
• Reading a value outside $[-1, 1]$ — for the sine and the cosine no such value exists.