Radians and the unit circle

IB Mathematics: Analysis and Approaches HL· 15 min read

1. Radian Measure and Conversion★☆☆☆☆⏱ 5 min

A radian is a dimensionless angle unit defined by the ratio of arc length to the radius of a circle. It is the standard unit for advanced trigonometry and calculus, simplifying many key formulas that become messy with degrees.

📘 Definition

Radian

rad (often omitted)

One radian is the angle subtended at the center of a circle by an arc equal in length to the circle's radius. A full rotation is $2 π$ radians = $36 0^{\circ}$ .

Example:

A half rotation is $π$ radians = $18 0^{\circ}$ .

📐 Worked Example

Convert $13 5^{\circ}$ to radians, and convert $\frac{7 π}{4}$ radians to degrees.

1
The conversion rule is $18 0^{\circ} = π$ radians. For degrees to radians, multiply by $\frac{π}{180}$ :
2
$135 \times \frac{\pi}{180} = \frac{3\pi}{4}$
3
For radians to degrees, multiply by $\frac{18 0 ^{\circ}}{π}$ :
4
$\frac{7\pi}{4} \times \frac{180^\circ}{\pi} = 315^\circ$

Exam tip:

Check the question to confirm which unit your final answer needs to be in.

2. Arc Length and Sector Area★★☆☆☆⏱ 5 min

When angles are measured in radians, formulas for arc length and sector area are much simpler than when using degrees. Both formulas depend on the circle radius $r$ and the central angle $θ$ .

📘 Definition

Arc Length

s

For a circle of radius $r$ , the length of an arc bounded by a central angle $θ$ (in radians) is $s = r θ$ .

Example:

For $r = 1$ and $θ = 2 π$ , $s = 2 π$ , which matches the circumference of a unit circle.

The area of a sector bounded by angle $θ$ (in radians) is $A = \frac{1}{2} r^{2} θ$ . This simplifies from the fraction-of-full-circle formula used with degrees.

📐 Worked Example

A sector of a circle with radius 6 cm has central angle $\frac{2 π}{3}$ radians. Find the arc length and area of the sector.

1
Use the arc length formula $s = r θ$ :
2
$s = 6 \times \frac{2\pi}{3} = 4\pi \text{ cm}$
3
Use the sector area formula $A = \frac{1}{2} r^{2} θ$ :
4
$A = \frac{1}{2} \times 6^2 \times \frac{2\pi}{3} = 12\pi \text{ cm}^2$

3. Trigonometric Functions on the Unit Circle★★☆☆☆⏱ 5 min

The unit circle extends trigonometric ratios from only acute angles in right triangles to all real angle measures. It is the foundational tool for working with trigonometric functions.

📘 Definition

Unit Circle Definition

For any angle $θ$ in standard position (counterclockwise from the positive x-axis), the point of intersection of the terminal side with the unit circle is $(x, y) = (cos θ, sin θ)$ , and $tan θ = \frac{y}{x} = \frac{s i n θ}{c o s θ}$ .

📐 Worked Example

Find the exact values of $sin (\frac{5 π}{6})$ , $cos (\frac{5 π}{6})$ and $tan (\frac{5 π}{6})$ using the unit circle.

1
$\frac{5 π}{6}$ is in Quadrant II, where only sine is positive.
2
The reference angle is $π - \frac{5 π}{6} = \frac{π}{6}$ , which has unit circle coordinates $(\frac{3}{2}, \frac{1}{2})$ .
3
Adjust the signs for Quadrant II: the coordinates are $(- \frac{3}{2}, \frac{1}{2})$ .
4
$sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}, \quad cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}, \quad tan\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{3}$

4. Exact Values for Common Angles★★★☆☆⏱ 5 min

IB exams regularly require exact values of trigonometric functions for common angles that are multiples of $\frac{π}{6}$ and $\frac{π}{4}$ . You are expected to recall these from the unit circle without a calculator.

✓ Quick check

Test your basic recall:

What is the value of $cos (\frac{π}{2})$ ?
- 0
- 1
- -1
- undefined
Reveal answer
0 —
Correct! At $\frac{π}{2}$ , the unit point is $(0, 1)$ , so $cos (\frac{π}{2}) = x = 0$ .
What is the value of $sin (\frac{3 π}{2})$ ?
- 0
- 1
- -1
- $\frac{2}{2}$
Reveal answer
-1 —
Correct! At $\frac{3 π}{2}$ , the unit point is $(0, - 1)$ , so $sin (\frac{3 π}{2}) = y = - 1$ .

📐 Worked Example

Find all $θ$ for $0 \leq θ < 2 π$ such that $sin θ = - \frac{3}{2}$ .

1
Sine is negative in Quadrants III and IV, so there are two solutions.
2
The reference angle for $sin θ = \frac{3}{2}$ is $\frac{π}{3}$ .
3
Quadrant III solution: $π + \frac{π}{3} = \frac{4 π}{3}$
4
Quadrant IV solution: $2 π - \frac{π}{3} = \frac{5 π}{3}$
5
Final solutions: $θ = \frac{4 π}{3}, \frac{5 π}{3}$

5. Common Pitfalls

Wrong move:

Using degrees instead of radians in arc length/sector area formulas

Why:

The formulas $s = r θ$ and $A = \frac{1}{2} r^{2} θ$ are only valid for radians. Using degrees gives an incorrect result.

Correct move:

Always convert degree measures to radians first by multiplying by $\frac{π}{180}$ .

Wrong move:

Swapping $sin θ$ and $cos θ$ on the unit circle

Why:

It is easy to mix up the x and y coordinates, leading to wrong exact values.

Correct move:

Remember: $(cos θ, sin θ) = (x, y)$ — alphabetical order, cos before sin, x before y.

Wrong move:

Forgetting to add the correct sign based on quadrant

Why:

Many students only recall the positive reference angle value and ignore the quadrant rule.

Correct move:

Always check the quadrant first, then apply the ASTC rule to get the correct sign.

Wrong move:

Adding a degree symbol to radian measures

Why:

This is a common formatting error that can cost you marks in IB exams.

Correct move:

Omit the degree symbol for radians: write $\frac{3 π}{4}$ , not $\frac{3 π}{4}^{\circ}$ .

6. Quick Reference Cheatsheet

Angle (°)	Angle (rad)	sinθ	cosθ	tanθ
0	0	0	1	0
30	$\frac{π}{6}$	$\frac{1}{2}$	$\frac{3}{2}$	$\frac{1}{3}$
45	$\frac{π}{4}$	$\frac{2}{2}$	$\frac{2}{2}$	$1$
60	$\frac{π}{3}$	$\frac{3}{2}$	$\frac{1}{2}$	$3$
90	$\frac{π}{2}$	$1$	$0$	undefined
180	$π$	$0$	$- 1$	$0$
270	$\frac{3 π}{2}$	$- 1$	$0$	undefined
360	$2 π$	$0$	$1$	$0$
Formulas		$s = r θ$	$A = \frac{1}{2} r^{2} θ$	$(x, y) = (cos θ, sin θ)$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Convert angle, find sector area
2022 · 2
Exact trig value on unit circle
2023 · 1
Arc length application problem

Going deeper

IB SyllabusIB AA HL Trigonometry Syllabus

What's Next

Radians and the unit circle are the absolute foundation for all further trigonometry in IB AA HL. This topic appears in every subsequent trigonometry unit, from graphing trigonometric functions, to proving and using trigonometric identities, to solving trigonometric equations, and even to representing complex numbers in polar form. Memorizing exact values for common angles and becoming comfortable with radian measure will save you time and prevent easy mistakes on nearly every trigonometry question you encounter in the final exam.