B. Trigonometric Functions · Radian Measure and Trigonometric Functions

Converting Between Radians and Degrees

B1.1 B1.2About 65 minutes

Learning goals

I can convert an angle from degrees to radians and from radians to degrees.
I can write a radian measure exactly in terms of π and approximately as a decimal.
I can draw angles in radians in standard position, including negative angles.
I can find angles coterminal with a given angle in radians.

Words to know

conversion factor
A fraction equal to 1 used to change units. Since π rad = 180°, both π/180° and 180°/π equal 1.

exact value (of an angle)
A radian measure written with π, such as 5π/6. A decimal such as 2.62 is an approximation.

standard position
An angle with its vertex at the origin and its initial arm on the positive x-axis.

terminal arm
The arm of an angle in standard position that rotates. It ends where the angle ends.

coterminal angles
Angles in standard position that share the same terminal arm. They differ by a multiple of 2π: θ + 2πn, where n is an integer.

One fact does all the work

Last lesson you found that a half turn is π radians. So

π rad = 180°

Every conversion comes from this single fact.

Degrees → radians: multiply by π/180°.
Radians → degrees: multiply by 180°/π.

A quick way to remember which one: the unit you want to end up with goes on the top of the fraction, so the old unit cancels.

Worked example

Example 1: Degrees to exact radians

Convert 150° to radians. Give the exact value and a decimal.

Multiply by π/180°: 150° × π/180° = 150π/180.
Simplify the fraction 150/180 = 5/6.
So 150° = 5π/6. As a decimal, 5π/6 ≈ 5(3.1416)/6 ≈ 2.62.

150° = 5π/6 ≈ 2.62 rad

Worked example

Example 2: Radians to degrees

Convert 7π/4 to degrees.

Multiply by 180°/π: (7π/4)(180°/π).
The π cancels: 7 × 180° ÷ 4 = 7 × 45° = 315°.

7π/4 = 315°. Shortcut: replace π with 180° and simplify.

Worked example

Example 3: A radian measure with no π

Convert 2.5 rad to degrees, to the nearest tenth.

There is no π to cancel, so use the calculator: 2.5 × 180 ÷ π.
2.5 × 57.2958... ≈ 143.2.

2.5 rad ≈ 143.2°

Two ways to write a radian measure

The curriculum asks you to write radian measures two ways:

In terms of π (exact): 40° = 40π/180 = 2π/9.
As a decimal, a rational number approximation: 2π/9 ≈ 0.70.

Use the π form for the special angles (multiples of π/6 and π/4) and in exact work. Use decimals when you are measuring, graphing on a decimal axis or using a model from real data. It is worth knowing a few decimals by heart: π/6 ≈ 0.52, π/4 ≈ 0.79, π/3 ≈ 1.05, π/2 ≈ 1.57, π ≈ 3.14, 3π/2 ≈ 4.71, 2π ≈ 6.28.

A double scale for one full turn. Every 30° step is π/6. Notice that 90° is 3π/6 = π/2 and 180° is 6π/6 = π: always simplify the fraction.

Degrees	0°	30°	45°	60°	90°	120°	135°	150°	180°
Radians (exact)	0	π/6	π/4	π/3	π/2	2π/3	3π/4	5π/6	π
Radians (decimal)	0	0.52	0.79	1.05	1.57	2.09	2.36	2.62	3.14

The special angles in the first two quadrants. For 180° to 360°, add π: for example 210° = π + π/6 = 7π/6.

Angles in standard position, in radians

Angles in standard position work the same way as in Grade 11, with radian labels:

The initial arm lies on the positive x-axis.
A positive angle turns counterclockwise; a negative angle turns clockwise.
The quadrant boundaries are at 0, π/2, π, 3π/2 and 2π.

Coterminal angles share a terminal arm. Adding or subtracting a full turn, 2π, gives another coterminal angle. All the angles coterminal with θ are θ + 2πn, where n is any integer.

The angle 3π/4 turns counterclockwise into quadrant II. The angle −π/3 turns clockwise into quadrant IV, and it shares its terminal arm with 5π/3, because −π/3 + 2π = 5π/3.

Worked example

Example 4: Coterminal angles

(a) Find one positive and one negative angle coterminal with 5π/6. (b) Find the angle between 0 and 2π that is coterminal with 23π/4.

(a) Add 2π = 12π/6: 5π/6 + 12π/6 = 17π/6.
Subtract 2π: 5π/6 − 12π/6 = −7π/6.
(b) 23π/4 is more than 2π = 8π/4. Subtract 2π twice (4π = 16π/4): 23π/4 − 16π/4 = 7π/4.
Check: 7π/4 is between 0 and 2π ✓ (it is 315°, in quadrant IV).

(a) 17π/6 and −7π/6 (other answers are possible). (b) 7π/4

Worked example

Example 5: Rotations per second to radians per second

A Ferris wheel makes one full turn every 40 s. Through what angle does it turn in 1 s and in 25 s?

One turn is 2π rad, so in 1 s it turns 2π ÷ 40 = π/20 rad (about 0.157 rad).
In 25 s: 25 × π/20 = 25π/20 = 5π/4.

π/20 rad each second; 5π/4 rad (225°) in 25 s. A rate like π/20 rad/s is called an angular speed. You will use it to build models in the lesson Modelling Periodic Phenomena.

Just for fun

Q: Why did the student bring a pie to trigonometry class? A: The teacher said a half turn is π radians, and she wanted to be prepared for the full turn.

Hands-on activity

Make a radian clock

You need: a paper plate (or a circle traced from a dinner plate onto paper), a ruler, a pencil and an analogue clock or a picture of one.

Draw a horizontal line and a vertical line through the centre of the plate (the x- and y-axes).
Each hour on a clock face is 1/12 of a turn: 2π ÷ 12 = π/6. Mark 12 equally spaced points around the rim, like the hours of a clock, starting with one point on the right end of the x-axis.
Starting at that point (where 3 o'clock would be) and going counterclockwise, label the points 0, π/6, π/3, π/2, ... up to 11π/6. Simplify every fraction.
In a second colour (or in brackets), write each angle in degrees.
Now add the multiples of π/4 between the points: π/4, 3π/4, 5π/4, 7π/4.
Test yourself: point to a mark and say its radian measure out loud. Then do it clockwise with negative angles: −π/6, −π/3, ...

What to look for: π/2 sits at the top (12 o'clock), π at the left (9 o'clock) and 3π/2 at the bottom (6 o'clock). Keep the plate: it is your unit circle for the next three lessons.

Your turn

Convert 60° to radians (exact).

Convert 225° to radians (exact).

Convert 5π/3 to degrees.

Convert 3π/2 to degrees.

Convert 1.2 rad to degrees, to the nearest tenth.

Convert 100° to radians, exactly and to two decimal places.

Find the angle between 0 and 2π that is coterminal with 11π/3.

More practice: warm-up

Convert 90° to radians.

Convert π to degrees.

Convert 45° to radians.

Convert 2π/3 to degrees.

Write π/6 as a decimal to two places.

More practice: on your own

Convert −135° to radians and describe the direction of rotation.

Convert 7π/6 to degrees and state the quadrant of its terminal arm.

Put these angles in order from smallest to largest: 2 rad, 100°, 3π/4, 1.8 rad.

A carousel turns once every 30 s. Find its angular speed in radians per second (exact), and the angle it turns in 20 s.

Find the angle between 0 and 2π that is coterminal with −5π/4, and name its quadrant.

Challenge

Earth turns once (2π rad) about every 24 hours relative to the Sun. Find this angular speed in rad/h. Then estimate how fast a point on the equator moves, using an equatorial radius of 6378 km.

Find, in radians, the smaller angle between the hands of a clock at (a) 4:00 and (b) 3:30.

List every angle coterminal with π/4 that lies strictly between −4π and 4π.

Check your understanding

Quiz

What is 120° in radians?

What is 5π/4 in degrees?

Which decimal is closest to 3π/2?

The terminal arm of −π/2 in standard position lies...

Which angle is coterminal with π/3?

To convert an angle from degrees to radians, multiply by...

For parents and teachers

Big idea (B1.1, B1.2): π radians = 180°. To convert, multiply by π/180° or by 180°/π. Exact radian measures keep π (5π/6); approximate ones are decimals (2.62).

How to help: Use any clock: each hour mark is π/6. Point to a time and ask for the angle from 3 o'clock, counterclockwise, in radians. Ask your teen to explain why 90° is π/2 (it is half of π, which is 180°).

Common difficulties:

Using the wrong conversion factor. Remind them the unit they want goes on top.
Not simplifying: 90π/180 should be π/2.
Writing "π = 180". π is about 3.14; π radians equals 180°.
Calculator in the wrong mode is not an issue for conversions, but it will be next lesson.