# Radians and the unit circle

> IB Mathematics: Analysis and Approaches HL · Geometry & Trigonometry
> Source: https://www.owlsprep.com/study/ib-math-aa-hl-u3-radians-and-the-unit-circle/

This sub-topic introduces radians as a natural angle unit, defines the unit circle, and connects unit circle coordinates to trigonometric ratios. You will learn to calculate exact trig values and solve arc and sector problems.

**Prerequisites:** [Basic right triangle trigonometry](https://www.owlsprep.com/study/ib-math-aa-hl-u2-right-triangle-trigonometry/)

## Learning objectives

- Convert between degree and radian angle measures
- Calculate arc length and sector area using radians
- Define sine, cosine, and tangent using the unit circle
- Find exact trigonometric values for common angles

## Radian Measure and Conversion

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.

**Radian** — 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 $2\pi$ radians = $360^\circ$.

*Notation:* rad (often omitted)

*Example:* A half rotation is $\pi$ radians = $180^\circ$.

**Worked example:** Convert $135^\circ$ to radians, and convert $\frac{7\pi}{4}$ radians to degrees.

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

> **tip**
>
> Unless asked for a decimal approximation, always leave radian measures in exact form in terms of $\pi$ for full marks.

> **Exam tip:** Check the question to confirm which unit your final answer needs to be in.

## Arc Length and Sector Area

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 $\theta$.

**Arc Length** — For a circle of radius $r$, the length of an arc bounded by a central angle $\theta$ (in radians) is $s = r\theta$.

*Notation:* s

*Example:* For $r=1$ and $\theta=2\pi$, $s=2\pi$, which matches the circumference of a unit circle.

The area of a sector bounded by angle $\theta$ (in radians) is $A = \frac{1}{2}r^2\theta$. 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 $\frac{2\pi}{3}$ radians. Find the arc length and area of the sector.

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

> **warning**
>
> These formulas only work if $\theta$ is in radians. Always convert to radians first if your angle is given in degrees.

## Trigonometric Functions on the Unit Circle

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.

**Unit Circle Definition** — For any angle $\theta$ in standard position (counterclockwise from the positive x-axis), the point of intersection of the terminal side with the unit circle is $(x,y) = (\cos\theta, \sin\theta)$, and $\tan\theta = \frac{y}{x} = \frac{\sin\theta}{\cos\theta}$.

**Worked example:** Find the exact values of $\sin\left(\frac{5\pi}{6}\right)$, $\cos\left(\frac{5\pi}{6}\right)$ and $\tan\left(\frac{5\pi}{6}\right)$ using the unit circle.

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

> **ASTC Rule**
>
> All Students Take Calculus: All positive (QI), Sine positive (QII), Tangent positive (QIII), Cosine positive (QIV).

## Exact Values for Common Angles

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

**Check your understanding**

Test your basic recall:

1. What is the value of $\cos\left(\frac{\pi}{2}\right)$?

   - 0
   - 1
   - -1
   - undefined

   *Why:* Correct! At $\frac{\pi}{2}$, the unit point is $(0,1)$, so $\cos\left(\frac{\pi}{2}\right) = x = 0$.

2. What is the value of $\sin\left(\frac{3\pi}{2}\right)$?

   - 0
   - 1
   - -1
   - $\frac{\sqrt{2}}{2}$

   *Why:* Correct! At $\frac{3\pi}{2}$, the unit point is $(0,-1)$, so $\sin\left(\frac{3\pi}{2}\right) = y = -1$.

**Worked example:** Find all $\theta$ for $0 \leq \theta < 2\pi$ such that $\sin\theta = -\frac{\sqrt{3}}{2}$.

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

## Common pitfalls

- **Wrong:** Using degrees instead of radians in arc length/sector area formulas
  - Why it fails: The formulas $s=r\theta$ and $A=\frac{1}{2}r^2\theta$ are only valid for radians. Using degrees gives an incorrect result.
  - Correct: Always convert degree measures to radians first by multiplying by $\frac{\pi}{180}$.
- **Wrong:** Swapping $\sin\theta$ and $\cos\theta$ on the unit circle
  - Why it fails: It is easy to mix up the x and y coordinates, leading to wrong exact values.
  - Correct: Remember: $(\cos\theta, \sin\theta) = (x,y)$ — alphabetical order, cos before sin, x before y.
- **Wrong:** Forgetting to add the correct sign based on quadrant
  - Why it fails: Many students only recall the positive reference angle value and ignore the quadrant rule.
  - Correct: Always check the quadrant first, then apply the ASTC rule to get the correct sign.
- **Wrong:** Adding a degree symbol to radian measures
  - Why it fails: This is a common formatting error that can cost you marks in IB exams.
  - Correct: Omit the degree symbol for radians: write $\frac{3\pi}{4}$, not $\frac{3\pi}{4}^\circ$.

## Cheatsheet

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

## 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.

- [Right triangle trigonometry](https://www.owlsprep.com/study/ib-math-aa-hl-u3-right-triangle-trigonometry/)
- [Non-right triangle trigonometry](https://www.owlsprep.com/study/ib-math-aa-hl-u3-non-right-triangle-trigonometry/)
- [Trigonometric equations](https://www.owlsprep.com/study/ib-math-aa-hl-u3-trigonometric-equations/)

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