# Right triangle trigonometry and the unit circle

> IB Mathematics Analysis & Approaches SL · Geometry & Trigonometry
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u3-right-triangle-trigonometry-and-the/

This sub-topic introduces core trigonometric ratios for right-angled triangles, then extends these definitions to all real angles using the unit circle. You will learn to find exact values for common angles and solve basic right triangle problems.

**Prerequisites:** [Pythagoras' theorem](https://www.owlsprep.com/study/ib-math-aa-sl-u2-pythagoras-theorem/); Basic angle properties

## Learning objectives

- Define sine, cosine and tangent for acute angles in right triangles
- Extend trigonometric definitions to all angles using the unit circle
- Calculate exact trigonometric values for standard common angles
- Solve for unknown sides and angles in right triangles

## Trigonometric Ratios in Right Triangles

**Trigonometric Ratios** — For an acute angle $\theta$ in a right-angled triangle, each ratio is the quotient of two sides relative to $\theta$: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

*Notation:* $\sin \theta$, $\cos \theta$, $\tan \theta$

*Example:* For a 30° angle with opposite side 2 and hypotenuse 4, $\sin 30° = \frac{2}{4} = 0.5$

> **SOH-CAH-TOA**
>
> SOH = **S**ine = **O**pposite / **H**ypotenuse  
> CAH = **C**osine = **A**djacent / **H**ypotenuse  
> TOA = **T**angent = **O**pposite / **A**djacent

**Worked example:** A right triangle has hypotenuse 10 cm and an angle of 35°. Find the length of the side opposite the 35° angle, correct to 1 decimal place.

1. We know the hypotenuse and need the opposite side, so we use the sine ratio:
2. $$\sin 35^\circ = \frac{\text{opposite}}{10}$$
3. Rearrange to isolate the unknown opposite side:
4. $$\text{opposite} = 10 \times \sin 35^\circ$$
5. Calculate with a calculator (ensure it is set to degree mode):
6. $$\text{opposite} \approx 10 \times 0.5736 = 5.7 \text{ cm}$$

*Calculator:* allowed

## Special Right Triangles and Exact Values

The 30-60-90 and 45-45-90 special right triangles have simple whole-number side ratios that produce exact trigonometric values. These values are almost always required in non-calculator Paper 1 exam questions.

**Special Right Triangles** — Right triangles with angles that produce simple, memorable exact trigonometric values, with side ratios of $1:1:\sqrt{2}$ for 45-45-90 and $1:\sqrt{3}:2$ for 30-60-90.

**Worked example:** Find the exact value of $\tan 60°$.

1. Recall the 30-60-90 triangle side ratios: opposite 30° = 1, opposite 60° = $\sqrt{3}$, hypotenuse = 2.
2. For an angle of 60°, the opposite side is $\sqrt{3}$ and the adjacent side is 1.
3. $$\tan 60^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3}$$

**Check your understanding**

Test your knowledge of exact values

1. What is the exact value of $\sin 45°$?

   - $\frac{1}{2}$
   - $\frac{\sqrt{2}}{2}$
   - $\frac{\sqrt{3}}{2}$
   - $1$

   *Why:* Correct! For 45-45-90, opposite = 1, hypotenuse = $\sqrt{2}$, so $\sin 45° = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$ after rationalizing.

*Calculator:* forbidden

## Unit Circle Definition of Trigonometric Functions

Right triangle trigonometry only works for acute angles between 0° and 90°. To define trigonometric functions for any angle, positive or negative, we use the unit circle.

**Unit Circle Definition** — For any angle $\theta$ measured counterclockwise from the positive x-axis, the coordinates of the point where the terminal side of $\theta$ intersects the unit circle are $(\cos \theta, \sin \theta)$.

$$x^2 + y^2 = 1, \quad \cos\theta = x, \quad \sin\theta = y, \quad \tan\theta = \frac{y}{x} = \frac{\sin\theta}{\cos\theta}$$

**Worked example:** Find the value of $\cos 180°$ using the unit circle.

1. A 180° angle lies along the negative x-axis. Its intersection point with the unit circle is $(-1, 0)$.
2. By definition, the x-coordinate of the point equals $\cos \theta$.
3. $$\cos 180^\circ = -1$$

> **warning**
>
> Unlike right triangle ratios, unit circle trigonometric values can be negative. The sign depends on which quadrant the terminal side of the angle falls into.

*Calculator:* allowed

## Reference Angles for Exact Values

To find the exact value of a trigonometric function for any angle, we find the reference angle (the acute angle between the terminal side and the x-axis), get the magnitude from the special triangle, then apply the correct sign based on the quadrant.

> **All Students Take Calculus (ASTC)**
>
> Quadrant 1: **All** ratios are positive  
> Quadrant 2: **S**ine is positive  
> Quadrant 3: **T**angent is positive  
> Quadrant 4: **C**osine is positive

**Worked example:** Find the exact value of $\sin 150°$.

1. 150° is in Quadrant 2, where sine is positive per the ASTC rule.
2. Calculate the reference angle for Quadrant 2: $180° - 150° = 30°$.
3. We know $\sin 30° = \frac{1}{2}$, so the magnitude is $\frac{1}{2}$ with a positive sign.
4. $$\sin 150^\circ = +\frac{1}{2}$$

*Calculator:* forbidden

## Common pitfalls

- **Wrong:** Forgetting to check that your calculator is in degree mode
  - Why it fails: IB exams almost always use degrees for trigonometry, and radian mode will give incorrect results
  - Correct: Always confirm your calculator's mode before starting any trig calculation
- **Wrong:** Mixing up opposite and adjacent sides relative to the working angle
  - Why it fails: Labels are relative to the angle you are using, not fixed to the triangle
  - Correct: Always re-label sides for the specific angle you are working with
- **Wrong:** Forgetting to add the negative sign for values outside Quadrant 1
  - Why it fails: Students often remember the magnitude from the reference angle but ignore the sign rule
  - Correct: Always check the quadrant and apply the ASTC rule before writing your final answer
- **Wrong:** Claiming $\tan 90° = 0$
  - Why it fails: $\tan \theta = \frac{\sin \theta}{\cos \theta}$, and $\cos 90° = 0$, so division by zero is undefined
  - Correct: Recognize that tangent is undefined at 90°, 270°, and all coterminal angles

## Cheatsheet

| Angle (°) | $\sin \theta$ | $\cos \theta$ | $\tan \theta$ |
| --- | --- | --- | --- |
| 0 | 0 | 1 | 0 |
| 30 | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ |
| 45 | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | 1 |
| 60 | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
| 90 | 1 | 0 | Undefined |
| 180 | 0 | -1 | 0 |
| 270 | -1 | 0 | Undefined |
| 360 | 0 | 1 | 0 |

## What's next

Right triangle trigonometry and the unit circle are the foundation for all further trigonometry in IB AA SL. Understanding the definitions of sine, cosine and tangent from the unit circle will let you work with trigonometric identities, graphs, and equations, all of which are heavily tested in exams. Exact values from special triangles and the unit circle are almost always required in Paper 1 non-calculator questions, so mastering this topic early will save you time and points later. Next, you will build on these core definitions to explore more advanced trigonometric topics.

- [Sine rule, cosine rule and area of triangles](https://www.owlsprep.com/study/ib-math-aa-sl-u3-sine-rule-cosine-rule-and/)
- [Graphs of trigonometric functions](https://www.owlsprep.com/study/ib-math-aa-sl-u3-graphs-of-trigonometric-functions/)
- [Pythagorean and double angle trigonometric identities](https://www.owlsprep.com/study/ib-math-aa-sl-u3-pythagorean-and-double-angle-trigonometric/)

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