Plotting sine, cosine and tangent

From a table of angles to a graph

A table lists a few angles. A graph shows all of them. numpy makes the angles and the values in two lines:

• np.linspace(0, 2 * np.pi, 200) makes $200$ evenly spaced numbers from $0$ to $2\pi$. It includes both ends.
• np.sin(x) works on every number of the array at once. There is no loop in your code.
• plt.plot(x, y) joins the points with short straight lines. With $200$ points in a turn, the lines look like a smooth curve.

Sine and cosine

The sine and cosine are smooth waves. Both stay between $-1$ and $1$, and both repeat every $2\pi$. That length is the period. The cosine is the sine slid to the left by $\dfrac{\pi}{2}$:

• $\cos x = \sin\left(x + \dfrac{\pi}{2}\right)$.

So the two graphs have the same shape. The sine starts at $0$, and the cosine starts at $1$.

Amplitude, speed and phase shift

A general wave has three numbers that you can change:

• $y = A\sin\big(B(x - C)\big)$.

Read them one at a time.

• A, the amplitude, is the height of the peaks. The wave goes from $-A$ to $A$.
• B, the speed, sets how many cycles fit into $2\pi$. The period is $\dfrac{2\pi}{B}$.
• C, the phase shift, slides the whole wave to the right by $C$. A negative $C$ slides it to the left.

In numpy, this is A * np.sin(B * (x - C)).

Practise with new numbers

The tangent graph

The tangent is $\tan x = \dfrac{\sin x}{\cos x}$. At $x = \dfrac{\pi}{2}$, $\dfrac{3\pi}{2}$ and every step of $\pi$ from them, the cosine is zero. The tangent is undefined there, and its graph has vertical asymptotes, lines it comes close to but never touches. The graph repeats every $\pi$, not every $2\pi$.

A computer never gives infinity here. np.tan(np.pi / 2) is about $1.6 \times 10^{16}$, a huge number, because the stored $\dfrac{\pi}{2}$ is a tiny bit off. That causes two problems on a plot:

• the huge values squash the rest of the graph;
• plt.plot joins the last point before an asymptote to the first point after it with a steep line.

Fix both. Replace the huge values with np.nan, which means "not a number" and leaves a gap in the line, and fix the height of the picture with plt.ylim:

Plotting in degrees

If your angles are in degrees, convert them before the function and label the axis in degrees:

The usual slip is np.sin(d) with $d$ in degrees. That draws a crowd of tiny waves, because every degree is read as one radian.

Try it in Python

The program draws a wave with the numbers $A$, $B$ and $C$ beside the cosine, and the tangent with its asymptotes hidden.

Common mistakes

• Multiplying instead of dividing for the period. A bigger $B$ means a faster wave, so a shorter period: $\dfrac{2\pi}{B}$.
• Mixing up the shift. In $\sin\big(B(x - C)\big)$ the wave moves to the right by $C$, not to the left.
• Plotting the tangent without a guard. The steep joining lines hide the real shape.
• Degrees given to np.sin. Convert with np.radians first.
• Too few points. With only $10$ points in a turn, the wave looks like a polygon. Use at least $100$.

Remember these

• np.linspace(start, stop, n) makes $n$ evenly spaced numbers, both ends included.
• $y = A\sin\big(B(x - C)\big)$: amplitude $A$, period $\dfrac{2\pi}{B}$, shifted right by $C$.
• $\cos x = \sin\left(x + \dfrac{\pi}{2}\right)$. Sine and cosine repeat every $2\pi$. Tangent repeats every $\pi$.
• Hide the huge values of the tangent with np.nan, and set plt.ylim.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. For $y = 3\sin(2x)$, what are the amplitude and the period?

2. Which phase shift $C$ turns $\sin x$ into $\cos x$ in $A\sin\big(B(x - C)\big)$?

3. Why does np.tan(np.pi / 2) not give infinity?

4. A plot of the tangent shows steep vertical lines joining the branches. How do you remove them?

5. How many points does np.linspace(0, 2 * np.pi, 200) make, and how far apart are they?