# Unit circle, radian measure, trigonometric identities

> IB Mathematics: Analysis and Approaches HL · Geometry and trigonometry
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u3-unit-circle-radian-measure-trigonometric/

This subtopic introduces radian angle measure, the unit circle for evaluating trigonometric functions of all angles, and core trigonometric identities to simplify expressions and solve equations, forming the foundation for all further trigonometry work in IB AI HL.

**Prerequisites:** [Basic right-angled triangle trigonometry](https://www.owlsprep.com/study/ib-math-ai-hl-u2-right-triangle-trigonometry/); Degree measure for angles

## Learning objectives

- Convert between degree and radian measure for angles
- Use the unit circle to find exact trigonometric values for any angle
- Apply core Pythagorean and double-angle trigonometric identities
- Solve basic trigonometric problems using identity relationships

## 1. Radian Measure and the Unit Circle Definition

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.

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

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

*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.

> **info**
>
> Tangent is defined as \tan \theta = \frac{\sin \theta}{\cos \theta}, which equals the slope of the terminal side of \theta.

## 2. Finding Exact Trigonometric Values

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.

> **ASTC Sign Rule**
>
> All Students Take Calculus: All functions positive Q1, Sine only positive Q2, Tangent only positive Q3, Cosine only positive Q4

**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. $$\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}, \quad \cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$$

**Check your understanding**

Test your understanding of quadrant signs:

1. What is the sign of \tan(\frac{7\pi}{6})?

   - Positive
   - Negative
   - Zero
   - Undefined

   *Why:* \frac{7\pi}{6} is in quadrant 3, where tangent is positive by the ASTC rule.

## 3. Core Trigonometric Identities

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.

**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)

## Common pitfalls

- **Wrong:** Forgetting to adjust the sign of the trigonometric value based on quadrant
  - Why it fails: Unit circle coordinates have different signs in each quadrant, so just the reference angle value will be wrong
  - Correct: Always check the quadrant and apply the ASTC rule to get the correct sign
- **Wrong:** Reversing the degree-radian conversion factor
  - Why it fails: Multiplying degrees by \frac{180}{\pi} instead of \frac{\pi}{180} gives incorrect values
  - Correct: Remember: degrees → radians: multiply by \frac{\pi}{180}, radians → degrees: multiply by \frac{180}{\pi}
- **Wrong:** Writing \sin^2 \theta as \sin \theta^2
  - Why it fails: This is ambiguous: \sin(\theta^2) ≠ (\sin \theta)^2 = \sin^2 \theta
  - Correct: Always write the square after the function name for squared trigonometric values
- **Wrong:** Choosing the wrong double-angle identity for \cos 2\theta
  - Why it fails: This requires extra calculation to find an unknown trig value, increasing error risk
  - Correct: 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

## Cheatsheet

| Concept | Formula/Rule |
| --- | --- |
| Degree-radian conversion | $180^\circ = \pi$ radians |
| Unit circle point | $(\cos \theta, \sin \theta)$ |
| Core Pythagorean identity | $\sin^2 \theta + \cos^2 \theta = 1$ |
| Tangent definition | $\tan \theta = \frac{\sin \theta}{\cos \theta}$ |
| ASTC Quadrant Rule | Q1: All +, Q2: Sin +, Q3: Tan +, Q4: Cos + |
| Sine double angle | $\sin 2\theta = 2\sin \theta \cos \theta$ |
| Cosine double angle | $\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta -1 = 1 - 2\sin^2 \theta$ |

## 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.

- [Graphs of Trigonometric Functions](https://www.owlsprep.com/study/ib-math-ai-hl-u3-graphs-of-trigonometric-functions/)
- [Solving Trigonometric Equations](https://www.owlsprep.com/study/ib-math-ai-hl-u3-solving-trigonometric-equations/)
- [Vectors: basics, operations, dot product](https://www.owlsprep.com/study/ib-math-ai-hl-u3-vectors-basics-operations-dot-product/)

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