# Right triangle trigonometry

> IB Mathematics AI HL · Geometry and trigonometry
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u3-right-triangle-trigonometry/

This module covers core trigonometric ratios for right-angled triangles, the foundation for all further trigonometry in IB AI HL. You will learn to find unknown sides, angles, and solve applied problems involving elevation and depression.

**Prerequisites:** [Pythagorean theorem for right triangles](https://www.owlsprep.com/study/ib-math-ai-hl-pythagorean-theorem/); Basic angle geometry and parallel line properties

## Learning objectives

- Label sides of a right triangle correctly relative to an acute angle
- Apply sine, cosine and tangent ratios to find unknown sides
- Use inverse trigonometric functions to find unknown acute angles
- Solve applied problems involving angles of elevation and depression

## Sides and Trigonometric Ratios

**Trigonometric Ratios** — Ratios that relate the lengths of sides of a right triangle to an acute angle, always defined relative to the angle of interest $\theta$.

*Example:* Three core ratios for right triangles are sine, cosine, and tangent.

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

**Worked example:** Label the opposite, adjacent, and hypotenuse sides for $\theta = 40^\circ$ at vertex A in a right triangle with right angle at C.

1. Step 1: The hypotenuse is always opposite the right angle, so side AB (opposite C) is the hypotenuse.
2. Step 2: The opposite side to $\theta$ at A does not touch A, so this is side BC.
3. Step 3: The adjacent side touches $\theta$ at A and is not the hypotenuse, so this is side AC.

## Finding Unknown Sides

When you know one acute angle and one side, you can use the matching trigonometric ratio to solve for any unknown side. First label all sides, select the ratio that connects the known and unknown side, then rearrange to isolate the unknown.

**Worked example:** A right triangle has an acute angle of $25^\circ$ and hypotenuse 12 cm. Find the length of the side opposite the $25^\circ$ angle, correct to 3 significant figures.

1. We know the hypotenuse, need the opposite side. The matching ratio is sine:
2. $$\sin(25^\circ) = \frac{x}{12}$$
3. Rearrange to solve for $x$:
4. $$x = 12 \sin(25^\circ)$$
5. Calculate with calculator in degrees mode:
6. $$x \approx 12 \times 0.4226 = 5.07 \text{ cm}$$

**Check your understanding**

Which ratio do you use to find an unknown adjacent side if you know the hypotenuse?

1. Which ratio is correct?

   - Sine
   - Cosine
   - Tangent

   *Why:* Correct! Cosine is defined as adjacent over hypotenuse, so it connects these two sides.

*Calculator:* allowed

## Finding Unknown Angles

To find an unknown acute angle when you know two sides, you use inverse trigonometric functions. These take the ratio of sides and return the corresponding angle in degrees (or radians).

**Inverse Trigonometric Function** — A function that reverses the trigonometric ratio, returning the angle that corresponds to a given side ratio. Written as $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$ on most calculators.

**Worked example:** A right triangle has adjacent side 7 cm and opposite side 4 cm to an unknown acute angle $\theta$. Find $\theta$ correct to 1 decimal place.

1. We know opposite and adjacent, so we use tangent for the ratio:
2. $$\tan \theta = \frac{4}{7}$$
3. Take inverse tangent of both sides to isolate $\theta$:
4. $$\theta = \tan^{-1}\left(\frac{4}{7}\right)$$
5. Calculate with calculator in degrees mode:
6. $$\theta \approx 29.7^\circ$$

> **warning**
>
> Always confirm your calculator is set to degrees mode for IB trig problems, unless radians are explicitly requested. A wrong mode will give you an incorrect result.

*Calculator:* allowed

## Applied Problems: Elevation and Depression

Right triangle trigonometry is commonly used for problems involving lines of sight. An angle of elevation is the angle upwards from the horizontal to a high object, while an angle of depression is the angle downwards from the horizontal to a low object. By alternate interior angles, these angles are equal for parallel horizontal lines.

**Worked example:** A hiker standing 80 m horizontally from the base of a cliff measures an angle of elevation of $28^\circ$ to the top of the cliff. If the hiker's eye level is 1.7 m above ground, find the total height of the cliff.

1. Draw a right triangle with horizontal base 80 m, vertical height $h$ from eye level to the cliff top, and angle of elevation $28^\circ$.
2. We know adjacent (80 m), need opposite ($h$), so use tangent:
3. $$\tan(28^\circ) = \frac{h}{80}$$
4. Rearrange and calculate:
5. $$h = 80 \tan(28^\circ) \approx 42.5 \text{ m}$$
6. Add the eye level height to get total cliff height:
7. $$42.5 + 1.7 = 44.2 \text{ m}$$

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Mixing up opposite and adjacent sides when choosing a trig ratio
  - Why it fails: You labeled the sides relative to the wrong angle
  - Correct: Always label all three sides explicitly relative to the given angle before selecting your ratio.
- **Wrong:** Leaving calculator in radians mode for trig calculations
  - Why it fails: You forgot to check the mode after using it for another topic
  - Correct: Check your calculator mode before every trig calculation, and reset to degrees if needed.
- **Wrong:** Adjusting the angle of depression unnecessarily by adding it to 90°
  - Why it fails: Confusion over where the angle is measured from
  - Correct: Angle of depression equals the angle of elevation from the object by alternate angles, so use it directly in your ratio.
- **Wrong:** Forgetting to add or subtract base height offsets in applied problems
  - Why it fails: You only calculated the height from the reference point, not the total requested height
  - Correct: Always read the problem carefully to check for offsets like eye level, and add/subtract them to your calculated value.
- **Wrong:** Using inverse trigonometric functions to find unknown sides
  - Why it fails: Confused the case of unknown side vs unknown angle
  - Correct: Inverse trig is only for finding unknown angles; use standard trig ratios and rearrange to find unknown sides.

## Cheatsheet

| Ratio | Formula | Use Case |
| --- | --- | --- |
| $\sin \theta$ | $\frac{\text{opposite}}{\text{hypotenuse}}$ | Find opposite/hypotenuse when one is known |
| $\cos \theta$ | $\frac{\text{adjacent}}{\text{hypotenuse}}$ | Find adjacent/hypotenuse when one is known |
| $\tan \theta$ | $\frac{\text{opposite}}{\text{adjacent}}$ | Find opposite/adjacent when one is known |
| $\sin^{-1}$ | $\theta = \sin^{-1}\left(\frac{o}{h}\right)$ | Find $\theta$ when you know opposite and hypotenuse |
| $\cos^{-1}$ | $\theta = \cos^{-1}\left(\frac{a}{h}\right)$ | Find $\theta$ when you know adjacent and hypotenuse |
| $\tan^{-1}$ | $\theta = \tan^{-1}\left(\frac{o}{a}\right)$ | Find $\theta$ when you know opposite and adjacent |

## What's next

Right triangle trigonometry is the foundational building block for all further trigonometry in IB AI HL. The core ratio logic you learned here extends to non-right triangles, 3D geometry problems, and periodic modeling with trigonometric functions. Mastery of this topic will make all subsequent trigonometry much faster and less error-prone, as many exam problems include a right triangle component even in more advanced contexts. Next, you will build on this knowledge to solve problems that go beyond basic right triangles.

- [Non-right triangles: Law of Sines, Law of Cosines, area](https://www.owlsprep.com/study/ib-math-ai-hl-u3-non-right-triangles-law-of/)
- [Unit circle, radian measure, trigonometric identities](https://www.owlsprep.com/study/ib-math-ai-hl-u3-unit-circle-radian-measure-trigonometric/)
- [Graphs of trigonometric functions](https://www.owlsprep.com/study/ib-math-ai-hl-u3-graphs-of-trigonometric-functions/)

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