Degrees, radians and the unit circle

Two ways to measure a turn

You have measured angles in degrees since Class 6. A full turn is $360^\circ$, a half turn is $180^\circ$ and a right angle is $90^\circ$. Mathematicians and computers prefer another unit, the radian. A full turn is $2\pi$ radians, so a half turn is exactly $\pi$ radians.

• To change degrees to radians, multiply by $\dfrac{\pi}{180}$.
• To change radians to degrees, multiply by $\dfrac{180}{\pi}$.
• One radian is about $57.3^\circ$.

In Python, math.radians(180) gives 3.141592653589793, and math.degrees(math.pi / 2) gives 90.0. The number $\pi$ is stored as math.pi. Never type 3.14: it has only three digits, and the error shows up in the third decimal of every answer.

Python measures angles in radians

Every trigonometric function in Python expects radians. If you give it degrees, it does not complain. It just reads your number as radians and gives a wrong answer.

• math.sin(30) is about $-0.988$, because $30$ radians is about five turns, not $30^\circ$.
• math.sin(math.radians(30)) is the sine of $30^\circ$, about $0.5$.

The numpy versions work on whole arrays: np.sin, np.cos, np.tan, np.radians and np.degrees.

The unit circle

In the chapter Introduction to Trigonometry you found sine and cosine from a right triangle: $\sin A = \dfrac{\text{opposite}}{\text{hypotenuse}}$. Now draw a circle of radius $1$ with its centre at the origin. Turn a radius anticlockwise from the positive x-axis by an angle $\theta$. Its tip is a point $P$. Drop a perpendicular from $P$ to the x-axis.

The triangle has a hypotenuse of $1$. So the side along the axis is $\cos\theta$, and the upright side is $\sin\theta$. That gives the whole definition:

• the point at angle $\theta$ is $(\cos\theta, \sin\theta)$.

This works for every angle, also for angles above $90^\circ$, negative angles and angles above a full turn.

Read the signs from the picture. In each quadrant, ask on which side of the axes the point lies.

• Quadrant I, $0^\circ$ to $90^\circ$: cosine and sine are both positive.
• Quadrant II, $90^\circ$ to $180^\circ$: cosine is negative, sine is positive.
• Quadrant III, $180^\circ$ to $270^\circ$: both are negative.
• Quadrant IV, $270^\circ$ to $360^\circ$: cosine is positive, sine is negative.

The tangent is $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$. It is undefined where $\cos\theta = 0$, at $90^\circ$ and $270^\circ$.

An angle bigger than a turn lands on the same point as a smaller one. The angle $380^\circ$ is the same as $20^\circ$. In Python, 380 % 360 gives 20, and -60 % 360 gives 300, because % never returns a negative number for a positive divisor.

The six ratios in a loop

Besides sine, cosine and tangent, there are three reciprocal ratios: $\csc\theta = \dfrac{1}{\sin\theta}$, $\sec\theta = \dfrac{1}{\cos\theta}$ and $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$. Python has no math.sec, so you write 1 / math.cos(a).

A loop builds a table of the ratios of the standard angles:

The standard values you learnt in Class 10 appear: $\sin 30^\circ = 0.5$, $\sin 45^\circ = \dfrac{1}{\sqrt{2}} = 0.7071$ and $\sin 60^\circ = \dfrac{\sqrt{3}}{2} = 0.866$. Where a ratio is undefined, as $\tan 90^\circ$ is, a guard such as if abs(c) > 1e-9: keeps the program from dividing by a number that is nearly zero.

sin(pi) is not exactly zero

Run print(math.sin(math.pi)). It prints 1.2246467991473532e-16, not 0.0. The number $\pi$ cannot be stored exactly. The computer keeps about $16$ digits, so the answer carries a rounding error of about $10^{-16}$. The same happens in math.sin(math.radians(30)), which gives 0.49999999999999994, not 0.5.

So a test such as math.sin(math.radians(30)) == 0.5 is False. Compare with a tolerance instead:

• math.isclose(a, b) is true when $a$ and $b$ agree to about nine digits;
• to compare with zero, give an absolute tolerance, math.isclose(x, 0, abs_tol=1e-9), because a relative tolerance means nothing next to zero;
• for arrays, use np.isclose and np.allclose.

Practise with new numbers

Try it in Python

The program converts angles, prints a table of ratios in a loop, shows the rounding error in $\sin\pi$, and draws the unit circle with the point at $50^\circ$.

Common mistakes

• Degrees where radians are needed. math.sin(30) is not the sine of $30^\circ$. Convert with math.radians first.
• Typing 3.14 for $\pi$. Use math.pi or np.pi.
• Comparing floats with ==. Use math.isclose or np.allclose.
• Forgetting the quadrant signs. A point in quadrant III has a negative cosine and a negative sine.
• Writing sin without math. Python then says name 'sin' is not defined. Write math.sin, or import it with from math import sin.

Remember these

• $360^\circ = 2\pi$ rad. Degrees to radians: multiply by $\dfrac{\pi}{180}$, or call math.radians.
• Python's sin, cos and tan take radians.
• The point at angle $\theta$ on the unit circle is $(\cos\theta, \sin\theta)$. The signs follow the quadrants.
• Compare floats with math.isclose(a, b, abs_tol=...), never with ==.

Check yourself

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

1. Convert $135^\circ$ to radians, as a multiple of $\pi$.

2. What does math.cos(math.radians(120)) give, and in which quadrant does the point lie?

3. A student writes math.sin(90) and gets 0.894, not 1.0. What went wrong?

4. What does -30 % 360 give, and which angle on the circle is it?

5. Why does math.sin(math.pi) == 0 print False, and what should you write?