Right triangle trigonometry

IB Mathematics AI HL· 45 min read

1. Sides and Trigonometric Ratios★☆☆☆☆⏱ 10 min

📘 Definition

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 $θ$ .

Example:

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

📐 Worked Example

Label the opposite, adjacent, and hypotenuse sides for $θ = 4 0^{\circ}$ 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 $θ$ at A does not touch A, so this is side BC.
3
Step 3: The adjacent side touches $θ$ at A and is not the hypotenuse, so this is side AC.

2. Finding Unknown Sides★★☆☆☆⏱ 15 min

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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 $2 5^{\circ}$ and hypotenuse 12 cm. Find the length of the side opposite the $2 5^{\circ}$ 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
$x \approx 12 \times 0.4226 = 5.07 \text{ cm}$

✓ Quick check

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

Which ratio is correct?
- Sine
- Cosine
- Tangent
Reveal answer
Cosine —
Correct! Cosine is defined as adjacent over hypotenuse, so it connects these two sides.

3. Finding Unknown Angles★★☆☆☆⏱ 15 min

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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).

📘 Definition

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 $θ$ . Find $θ$ 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 $θ$ :
4
$\theta = \tan^{-1}\left(\frac{4}{7}\right)$
5
Calculate with calculator in degrees mode:
6
$\theta \approx 29.7^\circ$

4. Applied Problems: Elevation and Depression★★★☆☆⏱ 20 min

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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 $2 8^{\circ}$ 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 $2 8^{\circ}$ .
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
$42.5 + 1.7 = 44.2 \text{ m}$

5. Common Pitfalls

Wrong move:

Mixing up opposite and adjacent sides when choosing a trig ratio

Why:

You labeled the sides relative to the wrong angle

Correct move:

Always label all three sides explicitly relative to the given angle before selecting your ratio.

Wrong move:

Leaving calculator in radians mode for trig calculations

Why:

You forgot to check the mode after using it for another topic

Correct move:

Check your calculator mode before every trig calculation, and reset to degrees if needed.

Wrong move:

Adjusting the angle of depression unnecessarily by adding it to 90°

Why:

Confusion over where the angle is measured from

Correct move:

Angle of depression equals the angle of elevation from the object by alternate angles, so use it directly in your ratio.

Wrong move:

Forgetting to add or subtract base height offsets in applied problems

Why:

You only calculated the height from the reference point, not the total requested height

Correct move:

Always read the problem carefully to check for offsets like eye level, and add/subtract them to your calculated value.

Wrong move:

Using inverse trigonometric functions to find unknown sides

Why:

Confused the case of unknown side vs unknown angle

Correct move:

Inverse trig is only for finding unknown angles; use standard trig ratios and rearrange to find unknown sides.

6. Quick Reference Cheatsheet

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

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.

2025 · 1
Find unknown side of right triangle
2023 · 2
Solve elevation angle word problem

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.