Mathematics 9709 · Pure Mathematics 1

Radians and converting

A radian is another way to measure an angle, in place of degrees.
It is the unit that makes every circle formula in this chapter simple.

The whole chapter in one line. A radian measures an angle by its own arc, not by the number 360. Once you do that, the long $\tfrac{\theta}{360}\times2\pi r$ turns into a short $r\theta$ . Every formula here gets simpler.

The one fact you use all the time: $\pi\text{ rad}=180^\circ$ . To change units, multiply by $\tfrac{\pi}{180}$ (degrees to radians) or $\tfrac{180}{\pi}$ (radians to degrees).

What a radian actually is

Take a circle of radius $r$ . Go a distance $r$ along the edge. That is one arc as long as the radius.
The angle you have made is 1 radian. It is the angle whose arc is as long as the radius.

An arc as long as the radius makes an angle of 1 radian.

The whole edge $2\pi r$ is $2\pi$ radius-lengths of arc. So a full turn is $2\pi$ radians.
So half a turn ( $180^\circ$ ) is $\pi$ radians. This gives the link $\pi\text{ rad}=180^\circ$ between the two units.

Why a full turn is exactly 2π radians

One radian uses one radius-length of arc. The arc all the way round is the whole edge, $2\pi r$ . So it holds $2\pi$ of those radius-lengths. That means $2\pi$ radians. Take half of it. A half-turn of $180^\circ$ is $\pi$ radians. Everything else is built from this.

Converting both ways

It all comes from $\pi\text{ rad}=180^\circ$ . To change one unit into the other, multiply by the right fraction.

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

The two fractions are the same fraction turned upside down.
If you forget which one, write $\pi\text{ rad}=180^\circ$ at the side and cancel.

Worked example

Convert both ways: $30^\circ$ to radians, then $\tfrac{5\pi}{9}$ to degrees

Degrees to radians: multiply by $\tfrac{\pi}{180}$ . Cancel the common factor of 30:
$30\times\dfrac{\pi}{180}=\dfrac{30\pi}{180}=\dfrac{\pi}{6}$
Radians to degrees: multiply by $\tfrac{180}{\pi}$ . The $\pi$ cancels, so it is just arithmetic:
$\dfrac{5\pi}{9}\times\dfrac{180}{\pi}=\dfrac{5\times180}{9}=100^\circ$

Answer

30^\circ=\tfrac{\pi}{6}\text{ rad},\qquad \tfrac{5\pi}{9}\text{ rad}=100^\circ

A few angles are worth learning by heart as exact multiples of $\pi$ , because they come up all the time: $30^\circ=\tfrac{\pi}{6}$ , $45^\circ=\tfrac{\pi}{4}$ , $60^\circ=\tfrac{\pi}{3}$ , $90^\circ=\tfrac{\pi}{2}$ and $180^\circ=\pi$ .

Where the marks go

Common mistake

The arc and area formulas later in this chapter only work when

\theta

is in radians. If a question gives the angle in degrees, change it to radians first. This is the most common mistake in the whole topic.

Common mistake

When the question asks for radians “in terms of

\pi

”, leave the

\pi

in. Do not turn

\tfrac{\pi}{6}

into

0.524

Common mistake

Set your calculator to radian mode before you take

\sin

\cos

of a radian angle. In degree mode

\sin 0.6

gives the sine of

0.6^\circ

. That is a completely different number.

Now you try

Write $210^\circ$ in radians, in terms of $\pi$ ; then write $\tfrac{7\pi}{12}$ radians in degrees. (9709-style)

Your turn— tap to reveal the worked answer (multiply by π/180, then by 180/π)

Degrees to radians: $210\times\tfrac{\pi}{180}=\tfrac{210\pi}{180}=\tfrac{7\pi}{6}$ .

Radians to degrees: $\tfrac{7\pi}{12}\times\tfrac{180}{\pi}=\tfrac{7\times180}{12}=105$ .

Answer

210^\circ=\tfrac{7\pi}{6}\text{ rad},\quad \tfrac{7\pi}{12}\text{ rad}=105^\circ

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Radians and converting

A radian is another way to measure an angle, in place of degrees.
It is the unit that makes every circle formula in this chapter simple.

The one fact you use all the time: $\pi\text{ rad}=180^\circ$ . To change units, multiply by $\tfrac{\pi}{180}$ (degrees to radians) or $\tfrac{180}{\pi}$ (radians to degrees).

What a radian actually is

Take a circle of radius $r$ . Go a distance $r$ along the edge. That is one arc as long as the radius.
The angle you have made is 1 radian. It is the angle whose arc is as long as the radius.

An arc as long as the radius makes an angle of 1 radian.

The whole edge $2\pi r$ is $2\pi$ radius-lengths of arc. So a full turn is $2\pi$ radians.
So half a turn ( $180^\circ$ ) is $\pi$ radians. This gives the link $\pi\text{ rad}=180^\circ$ between the two units.

Why a full turn is exactly 2π radians

Converting both ways

It all comes from $\pi\text{ rad}=180^\circ$ . To change one unit into the other, multiply by the right fraction.

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

The two fractions are the same fraction turned upside down.
If you forget which one, write $\pi\text{ rad}=180^\circ$ at the side and cancel.

Worked example

Convert both ways: $30^\circ$ to radians, then $\tfrac{5\pi}{9}$ to degrees

Degrees to radians: multiply by $\tfrac{\pi}{180}$ . Cancel the common factor of 30:
$30\times\dfrac{\pi}{180}=\dfrac{30\pi}{180}=\dfrac{\pi}{6}$
Radians to degrees: multiply by $\tfrac{180}{\pi}$ . The $\pi$ cancels, so it is just arithmetic:
$\dfrac{5\pi}{9}\times\dfrac{180}{\pi}=\dfrac{5\times180}{9}=100^\circ$

Answer

30^\circ=\tfrac{\pi}{6}\text{ rad},\qquad \tfrac{5\pi}{9}\text{ rad}=100^\circ

Where the marks go

Common mistake

The arc and area formulas later in this chapter only work when

\theta

is in radians. If a question gives the angle in degrees, change it to radians first. This is the most common mistake in the whole topic.

Common mistake

When the question asks for radians “in terms of

\pi

”, leave the

\pi

in. Do not turn

\tfrac{\pi}{6}

into

0.524

Common mistake

Set your calculator to radian mode before you take

\sin

\cos

of a radian angle. In degree mode

\sin 0.6

gives the sine of

0.6^\circ

. That is a completely different number.

Now you try

Write $210^\circ$ in radians, in terms of $\pi$ ; then write $\tfrac{7\pi}{12}$ radians in degrees. (9709-style)

Your turn— tap to reveal the worked answer (multiply by π/180, then by 180/π)

Degrees to radians: $210\times\tfrac{\pi}{180}=\tfrac{210\pi}{180}=\tfrac{7\pi}{6}$ .

Radians to degrees: $\tfrac{7\pi}{12}\times\tfrac{180}{\pi}=\tfrac{7\times180}{12}=105$ .

Answer

210^\circ=\tfrac{7\pi}{6}\text{ rad},\quad \tfrac{7\pi}{12}\text{ rad}=105^\circ