Radian measure

What you will be able to do

• say what one radian is, and why a full turn is $2\pi$ radians;
• change any angle from degrees into radians, and back;
• know the radian measure of the standard angles.

Why another unit?

The number $360$ for a full turn is a choice made long ago, perhaps because a year has about $360$ days. Nothing in the circle itself gives that number. The circle does give one length for free: its radius. Measuring angles with the radius makes the formulas for arcs, areas and speeds simple. It is also the unit that calculus and Python use.

One radian

Take a circle of radius $r$, and lay a piece of string of length $r$ along the circle. The angle at the centre between the two ends is one radian, written $1$ rad, or $1^c$.

An arc of length $s$ holds $\dfrac{s}{r}$ such pieces of string. So the angle that it subtends at the centre is

The angle is a length divided by a length, so a radian has no unit of its own. The same angle gives the same $\frac{s}{r}$ in every circle. A bigger circle has a longer arc and a longer radius, in the same proportion.

A full turn is $2\pi$ radians

The whole circumference is $2\pi r$, and it belongs to a full turn. So

Halving both sides gives the key fact:

From it, $1$ rad $= \dfrac{180^\circ}{\pi}$, which is about $57.3^\circ$, a little less than $60^\circ$. In the other direction, $1^\circ = \dfrac{\pi}{180}$ rad, about $0.01745$ rad.

Changing the unit

• Degrees into radians: multiply by $\dfrac{\pi}{180}$.
• Radians into degrees: multiply by $\dfrac{180}{\pi}$.

Leave $\pi$ in the answer when the angle is a neat fraction of $\pi$.

Example 1. $40^\circ = 40 \times \dfrac{\pi}{180} = \dfrac{2\pi}{9}$ rad.

Example 2. $-\dfrac{7\pi}{4}$ rad $= -\dfrac{7\pi}{4} \times \dfrac{180}{\pi} = -315^\circ$.

Example 3. $3$ rad $= 3 \times \dfrac{180}{\pi} \approx 171.9^\circ$. With no $\pi$ in the radian measure, the degrees are not a whole number.

The standard angles come up so often that it pays to know them:

A quick way to find them: $30^\circ$ is $\frac{1}{6}$ of $180^\circ$, so it is $\frac{\pi}{6}$. Every standard angle is a simple fraction of $180^\circ$.

When no unit is written, an angle is in radians. So $\theta = 2$ means $2$ rad, about $114.6^\circ$. Write the degree sign whenever you mean degrees.

A worked example

The hour hand of a clock moves from 12 to 5. Through what angle does it turn, in degrees and in radians? Give the angle with its sign.

1. One hour mark. The $12$ marks share a full turn, so each is $\dfrac{360^\circ}{12} = 30^\circ$, that is $\dfrac{\pi}{6}$ rad.
2. Five marks. $5 \times 30^\circ = 150^\circ$, and $5 \times \dfrac{\pi}{6} = \dfrac{5\pi}{6}$ rad.
3. The sign. A clock's hands turn clockwise, so the angle is $-150^\circ$, or $-\dfrac{5\pi}{6}$ rad.

Check: $-\dfrac{5\pi}{6} \times \dfrac{180}{\pi} = -150^\circ$.

Practise with new numbers

Try it in Python

Python's trigonometry works in radians, so you need these changes whenever you use math.sin or math.cos. The program changes the standard angles into radians with the formula, and also writes each one as a multiple of $\pi$. Then it uses Python's own tools, math.radians and math.degrees.

Common mistakes

• Multiplying by the wrong fraction. From degrees to radians, multiply by $\frac{\pi}{180}$. If $90^\circ$ comes out as more than $5000$, you used $\frac{180}{\pi}$.
• Writing $\pi = 180$. $\pi$ is a number, about $3.1416$. It is $\pi$ radians that equals $180^\circ$.
• Giving degrees to Python. math.sin(30) reads $30$ as $30$ radians and gives $-0.988$, not $0.5$. Write math.sin(math.radians(30)).
• Dropping the degree sign. "$\theta = 45$" means $45$ radians, which is more than seven turns.

Remember these

• One radian is the angle at the centre when the arc is as long as the radius. In general $\theta = \frac{s}{r}$.
• A full turn is $2\pi$ rad, and $\pi$ rad $= 180^\circ$.
• Degrees into radians: multiply by $\frac{\pi}{180}$. Radians into degrees: multiply by $\frac{180}{\pi}$.
• An angle with no unit is in radians. Python's math functions use radians.

Check yourself

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

1. In your own words, what is one radian?

2. Change $210^\circ$ into radians.

3. Change $\dfrac{3\pi}{4}$ rad into degrees.

4. Which is bigger, $1$ radian or $60^\circ$?

5. Arun's program prints math.cos(60) and gets $-0.952$. What went wrong?