B. Trigonometric Functions · Radian Measure and Trigonometric Functions

What Is a Radian? Measuring Angles with Arc Length

B1.1About 65 minutes

Learning goals

I can explain what one radian is, using the arc length and the radius of a circle.
I can explain why one full rotation is 2π radians and a half rotation is π radians.
I can use θ = a/r and a = rθ to find an angle, an arc length or a radius.
I can estimate the size of an angle given in radians.

Words to know

radian
The angle at the centre of a circle that cuts off an arc exactly as long as the radius. One radian is about 57.3°.

arc length (a)
The distance along the curved part of a circle between two points on the circle.

central angle (subtended angle)
An angle whose vertex is the centre of the circle. We say the arc subtends this angle at the centre.

circumference
The distance all the way around a circle: C = 2πr.

rotation
A turn about a centre point. One full rotation is 360°, or 2π radians.

A more natural way to measure angles

Since elementary school you have measured angles in degrees: a full turn is 360°. That number is a human choice. It is handy because 360 divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and many more, but nothing about a circle forces it.

Mathematicians, physicists and engineers usually use a different unit, the radian, which comes straight from the circle itself. In this course, in Calculus and Vectors (MCV4U) and in every university math or science course you take, radians are the default. Formulas in calculus such as "the rate of change of sin x is cos x" are only true when x is in radians.

Remember from Grade 11: in Functions (MCR3U) you worked with angles in standard position from 0° to 360°, the CAST rule, related acute angles and the unit circle, all in degrees. In this unit you will redo all of that in radians and then graph the trigonometric functions with radian inputs.

The definition

Draw a circle with radius r. Take a piece of the circle (an arc) that is exactly as long as the radius. The angle this arc subtends at the centre is one radian (1 rad).

More generally, if an arc of length a subtends a central angle θ, then the angle in radians is

θ = a ÷ r (arc length divided by radius).

Rearranging gives the very useful arc length formula a = rθ, which works only when θ is in radians.

Because a and r are both lengths (for example, both in centimetres), the units cancel. A radian is really a pure number: "2.5 radians" means the arc is 2.5 radii long. That is why you will often see an angle written as just θ = 2.5, with no unit at all. An angle with no degree symbol is in radians.

The shaded sector has two radii and an arc, all of length r. The angle at the centre is 1 radian, a little less than 60°.

How many radians make a full turn?

The whole circle has circumference C = 2πr. So the angle for one complete rotation is

θ = a ÷ r = 2πr ÷ r = 2π radians.

That means 2π rad = 360° and π rad = 180°. Dividing, 1 rad = 180°/π ≈ 57.3°. Some other useful facts follow right away:

a half turn is π (about 3.14)
a quarter turn (a right angle) is π/2 (about 1.57)
three quarters of a turn is 3π/2 (about 4.71)

So an angle of 1 rad is acute, 2 rad and 3 rad are obtuse (between π/2 ≈ 1.57 and π ≈ 3.14), and 4, 5 and 6 rad are reflex angles (between π and 2π ≈ 6.28).

Six arcs, each one radius long, fit around the circle with a small piece left over. A full turn is 2π ≈ 6.28 radians.

Radians only!

Use a = rθ and θ = a/r only with θ in radians. If an angle is given in degrees, convert it first (next lesson), or use the fraction of the full turn: a = (θ°/360°) × 2πr.

Worked example

Example 1: Finding an angle from an arc

An arc of length 12 cm lies on a circle of radius 5 cm. Find the central angle in radians, and estimate it in degrees.

Use θ = a ÷ r = 12 ÷ 5 = 2.4.
The units (cm ÷ cm) cancel, so θ = 2.4 rad.
Estimate in degrees: 2.4 × 57.3° ≈ 137.5°. That is between 90° and 180°, so the angle is obtuse. Check: 2.4 is between π/2 ≈ 1.57 and π ≈ 3.14 ✓.

θ = 2.4 rad (about 137.5°).

Worked example

Example 2: Finding an arc length

A wheel has a radius of 30 cm. A point on its rim turns through 2.5 rad. How far does the point travel along the rim?

The angle is already in radians, so a = rθ.
a = 30 × 2.5 = 75.

The point travels 75 cm.

Worked example

Example 3: Exact radian measures

Find, in terms of π, the angle in radians for (a) a half turn, (b) a quarter turn, (c) one-sixth of a turn.

A full turn is 2π rad. Take the fraction of 2π.
(a) ½ × 2π = π.
(b) ¼ × 2π = π/2.
(c) ⅙ × 2π = 2π/6 = π/3.

(a) π, (b) π/2, (c) π/3. Leaving π in the answer keeps it exact.

Worked example

Example 4: A rolling bicycle wheel

A bicycle wheel has a radius of 0.35 m. The bike rolls 10 m without slipping. Through what angle does the wheel turn, in radians? How many revolutions is that?

When a wheel rolls without slipping, the distance rolled equals the arc length that touches the road.
θ = a ÷ r = 10 ÷ 0.35 ≈ 28.57 rad.
One revolution is 2π rad, so revolutions = 28.57 ÷ 2π ≈ 28.57 ÷ 6.283 ≈ 4.55.

About 28.6 rad, or about 4.55 revolutions.

Two circles, one twice as big as the other, with the same central angle. The arc doubles too, so arc ÷ radius stays 1.2 rad. The radian measure of an angle does not depend on the size of the circle.

Just for fun

Q: What did the degree say to the radian? A: "You're about 57 times bigger than me, and you still don't need a little circle on your head."

Hands-on activity

Wrap a radius around a can

You need: a round can, jar lid or plate, a piece of string or thin ribbon about 1 m long, a ruler, a pen or marker, and scissors (be careful with them).

Measure across the top of the can through its centre (the diameter) and halve it to get the radius r. Write it down.
Cut a piece of string exactly r long.
Make a starting mark on the rim. Lay the string along the rim from the mark and make a second mark at its end. The arc between the marks subtends 1 radian at the centre.
Keep going: lay the string from each new mark to the next, all the way around. Count the whole pieces that fit.
Estimate what fraction of a string-length is left over at the end.

What to look for: 6 whole lengths fit, with about 0.28 of a length left over, because a full turn is 2π ≈ 6.28 radians. The result is the same whether you use a small lid or a big plate. Bonus: hold the can's top over a protractor and check that one radian is a little under 60°.

Your turn

A circle has radius 4 cm. An arc of length 10 cm subtends what central angle, in radians?

A circle has radius 12 m. Find the length of the arc that subtends a central angle of 0.75 rad.

An arc of length 18 cm subtends 1.5 rad. Find the radius.

How many radians are in one full rotation? Give the exact value and a decimal.

Is an angle of 3 rad acute, obtuse or reflex? Explain.

A circle has radius 10 cm. Find the exact length of the arc that subtends π/3 rad, then give a decimal.

Explain why a radian has no real unit.

More practice: warm-up

About how many degrees is 1 radian?

How many radians is a half turn?

On a circle of radius 2 cm, an arc is 2 cm long. What is the central angle?

Use a = rθ with r = 3 and θ = 2.

How many radians is a quarter turn?

More practice: on your own

The minute hand of a wall clock is 9 cm long. Through what angle (in radians) does it turn in 20 minutes, and how far does its tip travel?

A car tire has a radius of 32 cm. It turns through 50 rad. How far does the car travel, in metres?

Which is larger: 2 rad or 120°? Explain.

A pendulum 80 cm long swings so that its tip moves along an arc of 30 cm. Find the angle of the swing in radians and in degrees.

Two circles have radii 3 cm and 6 cm. On each, a central angle of 1.2 rad cuts off an arc. Find both arc lengths. What does this show?

Challenge

Two towns lie due north and south of each other. Their latitudes differ by 0.1 rad. Using 6371 km for the radius of Earth, about how far apart are they along Earth's surface?

A sector is the "pizza slice" between two radii. Explain why its area is A = ½r²θ (θ in radians), then find the area when r = 6 cm and θ = 2/3 rad.

An arc is exactly as long as the circle's diameter. Find the central angle in radians and in degrees, and the fraction of the full circle it covers.

Check your understanding

Quiz

One radian is the central angle for which...

A full rotation equals how many radians?

A circle has radius 5 cm. What arc length subtends 3 rad?

About how many degrees is 1 radian?

An angle of 2 radians is...

The formula a = rθ works when θ is measured in...

For parents and teachers

Big idea (B1.1): a radian measures an angle by how many radii long its arc is: θ = arc length ÷ radius. A full turn is 2π radians (about 6.28), so 2π rad = 360° and π rad = 180°.

How to help: Do the string-and-can activity together; it makes "just over 6 radians in a circle" stick. Then quiz quickly: "Is 1 rad bigger or smaller than a right angle? Is 4 rad more or less than a half turn?" (smaller; more).

Common difficulties:

Using a = rθ with θ in degrees. The formula needs radians.
Thinking π means 180. π is a number (about 3.14); π radians is 180°.
Expecting a unit after a radian measure. An angle with no degree symbol is in radians.
Forgetting that a = rθ gives the curved distance, not the straight-line chord.