Unit circle, radian measure, trigonometric identities

IB Mathematics: Analysis and Approaches HL· 15 min read

1. 1. Radian Measure and the Unit Circle Definition★★☆☆☆⏱ 15 min

Radian measure is an alternative to degrees that simplifies all calculus and trigonometric calculations. One radian is the angle subtended at the center of a circle by an arc equal in length to the circle's radius.

📘 Definition

Radian Measure

r (radius), s (arc length), \theta (angle in radians)

For any circle, the angle \theta in radians is given by \theta = \frac{s}{r}

Example:

A full 360° rotation equals 2\pi radians, a half 180° rotation equals \pi radians

Use the relationship 180° = \pi radians to convert between the two units.

📐 Worked Example

Convert (a) 60° to radians, (b) \frac{3\pi}{4} radians to degrees.

1
Step 1: For degrees to radians, multiply by \frac{\pi}{180}:
2
$60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}$
3
Step 2: For radians to degrees, multiply by \frac{180}{\pi}:
4
$\frac{3\pi}{4} \times \frac{180}{\pi} = 135^\circ$

The unit circle is a circle of radius 1 centered at the origin (0,0). For any angle \theta measured counterclockwise from the positive x-axis, the intersection of the terminal side of \theta with the unit circle has coordinates (\cos \theta, \sin \theta). This extends trigonometric functions to all angles, not just acute angles.

2. 2. Finding Exact Trigonometric Values★★★☆☆⏱ 20 min

Exact values for common angles (0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} and their multiples) can be read directly from the unit circle. To find values for angles outside the first quadrant, you first find the reference angle (acute angle to the x-axis) then adjust the sign based on the quadrant.

📐 Worked Example

Find the exact values of \sin(\frac{5\pi}{6}) and \cos(\frac{5\pi}{6}).

1
Step 1: Identify the quadrant: \frac{\pi}{2} < \frac{5\pi}{6} < \pi, so quadrant 2.
2
Step 2: Calculate the reference angle: \pi - \frac{5\pi}{6} = \frac{\pi}{6}.
3
Step 3: Apply ASTC: sine is positive in Q2, cosine is negative. We know \sin(\frac{\pi}{6}) = \frac{1}{2} and \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}.
4
$\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}, \quad \cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$

✓ Quick check

Test your understanding of quadrant signs:

What is the sign of \tan(\frac{7\pi}{6})?
- Positive
- Negative
- Zero
- Undefined
Reveal answer
Positive —
\frac{7\pi}{6} is in quadrant 3, where tangent is positive by the ASTC rule.

3. 3. Core Trigonometric Identities★★★☆☆⏱ 25 min

Trigonometric identities are used to simplify expressions, solve equations, and rewrite trigonometric functions in more useful forms. The two most important sets of identities for this topic are Pythagorean identities and double-angle identities.

📘 Definition

Pythagorean Identity

Derived directly from the unit circle equation $x^{2} + y^{2} = 1$ , the core identity is:

Example:

Used to find \sin \theta from \cos \theta (or vice versa)

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7. 8. Common Pitfalls

Wrong move:

Forgetting to adjust the sign of the trigonometric value based on quadrant

Why:

Unit circle coordinates have different signs in each quadrant, so just the reference angle value will be wrong

Correct move:

Always check the quadrant and apply the ASTC rule to get the correct sign

Wrong move:

Reversing the degree-radian conversion factor

Why:

Multiplying degrees by \frac{180}{\pi} instead of \frac{\pi}{180} gives incorrect values

Correct move:

Remember: degrees → radians: multiply by \frac{\pi}{180}, radians → degrees: multiply by \frac{180}{\pi}

Wrong move:

Writing \sin^2 \theta as \sin \theta^2

Why:

This is ambiguous: \sin(\theta^2) ≠ (\sin \theta)^2 = \sin^2 \theta

Correct move:

Always write the square after the function name for squared trigonometric values

Wrong move:

Choosing the wrong double-angle identity for \cos 2\theta

Why:

This requires extra calculation to find an unknown trig value, increasing error risk

Correct move:

Pick the form matching your known value: use 1 - 2\sin^2 \theta if you know \sin \theta, 2\cos^2 \theta -1 if you know \cos \theta

9. Quick Reference Cheatsheet

Concept	Formula/Rule
Degree-radian conversion	$18 0^{\circ} = π$ radians
Unit circle point	$(cos θ, sin θ)$
Core Pythagorean identity	$sin^{2} θ + cos^{2} θ = 1$
Tangent definition	$tan θ = \frac{s i n θ}{c o s θ}$
ASTC Quadrant Rule	Q1: All +, Q2: Sin +, Q3: Tan +, Q4: Cos +
Sine double angle	$sin 2 θ = 2 sin θ cos θ$
Cosine double angle	$cos 2 θ = cos^{2} θ - sin^{2} θ = 2 cos^{2} θ - 1 = 1 - 2 sin^{2} θ$

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
Find exact trigonometric value
2022 · 2
Use identity to solve equation
2023 · 1
Convert between units of angle

Going deeper

syllabusOfficial IB AI HL SyllabusFull syllabus reference for this topic

What's Next

This subtopic forms the foundation for all further work with trigonometric functions, graphing, and equations in IB AI HL. You will use these identities to simplify complex expressions, solve trigonometric equations of all types, and model periodic phenomena like waves, seasonal cycles, and oscillations in applied contexts. Mastery of exact trig values from the unit circle will save you significant time on non-calculator Paper 1 questions, so it is worth memorizing these values early in your course.