Trigonometric Ratios

Q: How do you identify and label the three sides of a right-angled triangle relative to a reference angle \(\theta\)?

The Hypotenuse (\(H\)) is the longest side opposite the \(90^\circ\) angle, the Opposite (\(O\)) is opposite the angle \(\theta\), and the Adjacent (\(A\)) is next to \(\theta\).

Q: What mnemonic is used to remember the three trigonometric ratios?

SOH CAH TOA , which stands for \(\sin = \frac{O}{H}\), \(\cos = \frac{A}{H}\), and \(\tan = \frac{O}{A}\).

Q: Which calculator mode must be active when solving angle problems in degrees?

The calculator must be set to Degree Mode (indicated by a D or DEG on the screen).

Q: When should you use Pythagoras' Theorem instead of Trigonometric Ratios?

Use Pythagoras' Theorem when working purely with three side lengths, and Trigonometry when linking sides with angles.

Mathematics · Key Stage 3 (KS3) · 6 min read · Updated: Aug 27, 2026

Welcome to Trigonometric Ratios!

Have you ever wondered how builders calculate the exact angle of a roof, or how surveyors measure the height of a mountain without climbing to the top? They use a branch of mathematics called trigonometry, which literally means "triangle measuring".

In this chapter, we will learn how angles and side lengths are connected in right-angled triangles. Don't worry if this sounds tricky at first — we will break it down into simple, easy-to-follow steps!

1. The Anatomy of a Right-Angled Triangle

Before we can use any formulas, we must learn how to label the three sides of a right-angled triangle correctly. A right-angled triangle contains one interior angle of \(90^\circ\). The other two angles are acute (less than \(90^\circ\)).

We choose one of these acute angles as our reference angle, which we often label using the Greek letter theta: \(\theta\).

Labelling the Three Sides

Every side has a special name based on where it sits in relation to the right angle and our reference angle \(\theta\):

• Hypotenuse (\(H\)): This is always the longest side of the right-angled triangle. It sits directly opposite the \(90^\circ\) right angle.
• Opposite (\(O\)): This is the side directly opposite the reference angle \(\theta\) (across the triangle from it).
• Adjacent (\(A\)): The word adjacent means "next to". This is the side next to the reference angle \(\theta\), sitting between \(\theta\) and the right angle.

Top Tip: Always label the Hypotenuse first because its position never changes (it is always across from the \(90^\circ\) angle). Then find the angle \(\theta\) to label the Opposite and Adjacent sides!

Key Takeaway

The names Opposite and Adjacent change depending on which acute angle is chosen as \(\theta\). Never assume the bottom side is always the Adjacent side!

2. The Big Idea: Similar Triangles and Ratios

Why does trigonometry work? It is all based on similar triangles.

If two right-angled triangles have the exact same angle \(\theta\), they have the exact same shape — one is simply an enlarged version of the other. Because they are similar, the ratios between their side lengths remain constant, no matter how big or small the triangle is.

Did you know? If you draw a tiny right-angled triangle with an angle of \(30^\circ\) and a massive right-angled triangle with an angle of \(30^\circ\), the ratio of \(\frac{\text{Opposite}}{\text{Hypotenuse}}\) will be exactly \(0.5\) in both triangles!

3. The Three Primary Trigonometric Ratios

There are three main trigonometric ratios that link the angle \(\theta\) to pairs of sides:

1. Sine (written as \(\sin\))

\(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{O}{H}\)

2. Cosine (written as \(\cos\))

\(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{A}{H}\)

3. Tangent (written as \(\tan\))

\(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{O}{A}\)

Memory Aid: SOH CAH TOA

To remember these three important definitions, we use the famous mnemonic SOH CAH TOA:

• SOH: \(\mathbf{S}\text{in}(\theta) = \frac{\mathbf{O}\text{pposite}}{\mathbf{H}\text{ypotenuse}}\)
• CAH: \(\mathbf{C}\text{os}(\theta) = \frac{\mathbf{A}\text{djacent}}{\mathbf{H}\text{ypotenuse}}\)
• TOA: \(\mathbf{T}\text{an}(\theta) = \frac{\mathbf{O}\text{pposite}}{\mathbf{A}\text{djacent}}\)

4. Calculating Missing Side Lengths

To find a missing side length in a right-angled triangle, follow these four simple steps:

Step 1: Label the sides of the triangle (\(O\), \(A\), \(H\)) based on the given angle \(\theta\).
Step 2: Identify the side you know and the side you want to find.
Step 3: Pick the correct ratio from SOH CAH TOA that links these two sides.
Step 4: Substitute the values into the formula and solve the equation for the unknown side.

Case 1: The Unknown is on Top (Numerator)

Example: A triangle has an angle \(\theta = 30^\circ\), a Hypotenuse \(H = 10\text{ cm}\), and an unknown Opposite side \(x\).

1. We have \(H\) and want \(O\), so we use SOH: \(\sin(\theta) = \frac{O}{H}\)
2. Substitute the values: \(\sin(30^\circ) = \frac{x}{10}\)
3. Multiply both sides by \(10\): \(x = 10 \times \sin(30^\circ)\)
4. Calculate: \(x = 10 \times 0.5 = 5\text{ cm}\)

Case 2: The Unknown is on the Bottom (Denominator)

Example: A triangle has an angle \(\theta = 35^\circ\), an Opposite side \(O = 7\text{ cm}\), and an unknown Hypotenuse \(x\).

1. We have \(O\) and want \(H\), so we use SOH: \(\sin(\theta) = \frac{O}{H}\)
2. Substitute the values: \(\sin(35^\circ) = \frac{7}{x}\)
3. Rearrange carefully to solve for \(x\):
Multiply by \(x\): \(x \times \sin(35^\circ) = 7\)
Divide by \(\sin(35^\circ)\): \(x = \frac{7}{\sin(35^\circ)}\)
4. Calculate: \(x \approx 12.2\text{ cm}\) (rounded to \(1\) decimal place)

5. Calculating Missing Angles (Inverse Trigonometry)

When you know two side lengths and want to calculate a missing angle \(\theta\), use the inverse trigonometric functions:

• Inverse Sine: \(\theta = \sin^{-1}\left(\frac{O}{H}\right)\)
• Inverse Cosine: \(\theta = \cos^{-1}\left(\frac{A}{H}\right)\)
• Inverse Tangent: \(\theta = \tan^{-1}\left(\frac{O}{A}\right)\)

On your scientific calculator, you can access these inverse functions by pressing the SHIFT or 2ndF button followed by sin, cos, or tan.

Step-by-Step Angle Calculation

Example: A right-angled triangle has an Adjacent side \(A = 4\text{ cm}\) and a Hypotenuse \(H = 8\text{ cm}\). Find angle \(\theta\).

1. We know \(A\) and \(H\), so we choose CAH: \(\cos(\theta) = \frac{A}{H}\)
2. Set up the fraction: \(\cos(\theta) = \frac{4}{8} = 0.5\)
3. Apply the inverse function: \(\theta = \cos^{-1}(0.5)\)
4. On the calculator: press SHIFT \(\rightarrow\) cos \(\rightarrow\) \(0.5\) \(\rightarrow\) =
5. Result: \(\theta = 60^\circ\)

6. Essential Calculator Setup & Common Pitfalls

Calculator Mode Check

Always ensure your scientific calculator is set to Degree Mode (look for a small D or DEG at the top of the screen). If it shows R (Radians) or G (Gradians), your angle calculations will give incorrect answers!

Common Mistakes to Avoid

• Swapping Opposite and Adjacent: Remember that Opposite is directly across from the angle \(\theta\), while Adjacent is next to it.
• Division Errors with Fractions: If \(\sin(35^\circ) = \frac{7}{x}\), do not write \(x = 7 \times \sin(35^\circ)\). The correct rearrangement is \(x = \frac{7}{\sin(35^\circ)}\).
• Treating \(\sin\), \(\cos\), \(\tan\) as Numbers: You cannot separate \(\sin\) from its angle by dividing. Writing just "\(\sin\)" without an angle has no mathematical meaning — it is an operation, not a multiplying number.
• Confusing Inverse Notation: The symbol \(\sin^{-1}(x)\) means the inverse angle function. It does not mean \(\frac{1}{\sin(x)}\).
• Trig vs Pythagoras' Theorem: Use Pythagoras' Theorem when you are working strictly with three side lengths and no angles. Use Trigonometric Ratios when you are dealing with a mixture of side lengths and angles!

Quick Summary Checklist

• Hypotenuse (\(H\)) = opposite \(90^\circ\) angle (longest side).
• Opposite (\(O\)) = opposite the reference angle \(\theta\).
• Adjacent (\(A\)) = next to the reference angle \(\theta\).
• SOH: \(\sin(\theta) = \frac{O}{H}\)
• CAH: \(\cos(\theta) = \frac{A}{H}\)
• TOA: \(\tan(\theta) = \frac{O}{A}\)
• To find a missing angle, use inverse functions: \(\sin^{-1}\), \(\cos^{-1}\), \(\tan^{-1}\).
• Always check that your calculator display shows DEG or D.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

How do you identify and label the three sides of a right-angled triangle relative to a reference angle \(\theta\)?

The Hypotenuse (\(H\)) is the longest side opposite the \(90^\circ\) angle, the Opposite (\(O\)) is opposite the angle \(\theta\), and the Adjacent (\(A\)) is next to \(\theta\).

Why do trigonometric ratios stay constant for a given angle regardless of triangle size?

Because right-angled triangles sharing the same angle \(\theta\) are similar triangles, meaning their corresponding side lengths scale proportionally.

What are the definitions of the three primary trigonometric ratios?

\(\sin(\theta) = \frac{O}{H}\), \(\cos(\theta) = \frac{A}{H}\), and \(\tan(\theta) = \frac{O}{A}\).

What mnemonic is used to remember the three trigonometric ratios?

SOH CAH TOA, which stands for \(\sin = \frac{O}{H}\), \(\cos = \frac{A}{H}\), and \(\tan = \frac{O}{A}\).

How do you find a missing side length when the unknown is in the numerator of the ratio?

Multiply both sides of the trigonometric equation by the known denominator (e.g., \(x = H \times \sin(\theta)\)).

How do you calculate a missing side length when the unknown is in the denominator?

Rearrange by multiplying by the unknown and dividing by the trigonometric value (e.g., \(x = \frac{O}{\sin(\theta)}\)).

How do you calculate a missing angle in a right-angled triangle when two sides are known?

Use inverse trigonometric functions (\(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\)) on the ratio of the two known sides.

Which calculator mode must be active when solving angle problems in degrees?

The calculator must be set to Degree Mode (indicated by a D or DEG on the screen).

When should you use Pythagoras' Theorem instead of Trigonometric Ratios?

Use Pythagoras' Theorem when working purely with three side lengths, and Trigonometry when linking sides with angles.

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