Right-angled triangle trigonometry

IB Mathematics: Applications and Interpretation SL· 15 min read

1. 1. Identifying sides of a right-angled triangle★☆☆☆☆⏱ 10 min

📘 Definition

Sides of a right-angled triangle

All right-angled triangles have three sides, defined relative to any given acute angle $θ$ : the hypotenuse is always the longest side, opposite the right angle. The opposite side is across from $θ$ , and the adjacent side is next to $θ$ (not the hypotenuse).

Example:

For triangle $A B C$ right-angled at $C$ , with $θ$ at $A$ : hypotenuse = $A B$ , opposite = $B C$ , adjacent = $A C$

📐 Worked Example

In right-angled triangle $X Y Z$ , right-angled at $Y$ , with acute angle $3 0^{\circ}$ at $X$ . Identify hypotenuse, opposite and adjacent sides relative to the $3 0^{\circ}$ angle.

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Step 1: Locate the right angle at $Y$ . The hypotenuse is always opposite the right angle, so hypotenuse = $X Z$ .
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Step 2: The given angle is at $X$ , so the side across from $X$ is $Y Z$ . This is the opposite side.
3
Step 3: The remaining side next to $X$ (not the hypotenuse) is $X Y$ . This is the adjacent side.

2. 2. Trigonometric ratios and SOHCAHTOA★★☆☆☆⏱ 15 min

✓ Calculator OK

📘 Definition

Trigonometric ratios

For acute angle $θ$ :

The three core trigonometric ratios are defined as ratios of side lengths of a right-angled triangle:

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}

📐 Worked Example

A right-angled 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 2 decimal places.

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Step 1: We know the hypotenuse and need the opposite side. This matches the sine ratio from SOHCAHTOA.
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Step 2: Write the sine ratio, let the unknown opposite side be $x$ :
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$\sin(25^\circ) = \frac{x}{12}$
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Step 3: Rearrange to isolate $x$ :
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$x = 12 \times \sin(25^\circ)$
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Step 4: Calculate with a calculator in degree mode: $x \approx 12 \times 0.4226 = 5.07$ cm

Exam tip:

Always confirm your calculator is in degree mode before starting any trigonometry calculation for IB exams.

3. 3. Calculating unknown acute angles★★☆☆☆⏱ 15 min

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When you know two side lengths of a right-angled triangle, you can calculate an unknown acute angle using inverse trigonometric functions. These functions reverse the action of sine, cosine and tangent, returning the angle value from a given ratio.

📘 Definition

Inverse trigonometric functions

$θ = sin^{- 1} (k), θ = cos^{- 1} (k), θ = tan^{- 1} (k)$

If $sin θ = k$ , then $θ = sin^{- 1} (k)$ gives the angle whose sine ratio equals $k$ . This works identically for inverse cosine and tangent.

📐 Worked Example

Find the unknown acute angle $θ$ in a right-angled triangle, where the side adjacent to $θ$ is 7 cm and the hypotenuse is 10 cm. Give your answer to the nearest degree.

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Step 1: We know adjacent and hypotenuse, so we use the cosine ratio.
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Step 2: Write the ratio:
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$\cos\theta = \frac{7}{10} = 0.7$
4
Step 3: Apply inverse cosine to both sides to isolate $θ$ :
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$\theta = \cos^{-1}(0.7)$
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Step 4: Calculate with a calculator: $θ \approx 4 6^{\circ}$

4. 4. Applied right-angled trigonometry★★★☆☆⏱ 20 min

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Right-angled trigonometry is commonly used to solve real-world problems including finding heights of objects, distances between points, angles of elevation/depression, and bearings. Always start by drawing a labelled diagram to model the problem.

📐 Worked Example

The angle of elevation from a point 50 m horizontally from the base of a tree to the top of the tree is $2 2^{\circ}$ . Find the height of the tree, correct to 1 decimal place.

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Step 1: Draw a right-angled triangle: the horizontal distance (50 m) is adjacent to the $2 2^{\circ}$ angle, and the height $h$ of the tree is opposite the angle.
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Step 2: We have adjacent, need opposite, so use tangent:
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$\tan(22^\circ) = \frac{h}{50}$
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Step 3: Rearrange for $h$ :
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$h = 50 \times \tan(22^\circ)$
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Step 4: Calculate: $h \approx 50 \times 0.4040 = 20.2$ m

Exam tip:

Always label your diagram clearly with all given information to avoid mixing up opposite and adjacent sides.

5. Common Pitfalls

Wrong move:

Mixing up opposite and adjacent sides relative to the given angle

Why:

Opposite and adjacent are defined relative to the acute angle you are working with, not the right angle

Correct move:

Always re-label the triangle from the perspective of the angle you are using

Wrong move:

Leaving your calculator in radian mode instead of degree mode

Why:

IB AI SL almost exclusively uses degrees for trigonometry questions, leading to incorrect answers

Correct move:

Check your calculator mode before starting any trigonometry calculation

Wrong move:

Using the regular trigonometric function instead of inverse when finding an angle

Why:

A regular trig function takes an angle and gives a ratio; you need the inverse to get an angle from a ratio

Correct move:

Always use $sin^{- 1}$ , $cos^{- 1}$ or $tan^{- 1}$ when calculating an unknown angle

Wrong move:

Using Pythagoras' theorem when an angle is given

Why:

Pythagoras only works when you know two sides and have no angle information

Correct move:

If you have an angle, always use trigonometric ratios to find the unknown side

6. Quick Reference Cheatsheet

Concept	Rule	Notation
Hypotenuse	Longest side, opposite right angle	$c$
Opposite side	Side across from angle $θ$	$O$
Adjacent side	Side next to angle $θ$	$A$
Sine	SOH = $O / H$	$sin θ = O / H$
Cosine	CAH = $A / H$	$cos θ = A / H$
Tangent	TOA = $O / A$	$tan θ = O / A$
Find angle	Use inverse trig function	$θ = sin^{- 1} (O / H)$
Find side	Rearrange ratio to isolate unknown	$O = H \times sin θ$

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.

2022 · 1
Find missing side of right triangle
2023 · 2
Bearings problem with right triangle

What's Next

Right-angled triangle trigonometry is the foundation for all further trigonometry in IB AI SL. Next, you will extend these ideas to solve non-right angled triangles using the sine and cosine rules, which are required for most irregular triangle problems in exams. You will also use trigonometry to model periodic phenomena like tide heights and seasonal temperatures later in the course, so mastering SOHCAHTOA is critical for success in these more advanced topics.