O Level Trigonometry Worked Examples for 2026/2027 (Singapore MOE Syllabus) — Step-by-Step Worked Examples

Quick answer

When you see a trigonometry question in your O Level Additional Mathematics exam and feel like freezing, remember this: you probably already know the concept. It's just wrapped differently. I’ll walk you through four worked examples, step by step, to show you how to approach these questions calmly and gain those marks you deserve.

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What you need to know

Trigonometry is about understanding the relationships between the angles and sides of triangles. In O Level Additional Mathematics, you’ll use trigonometric ratios like sine, cosine, and tangent to solve problems. These are often applied in the context of right-angled triangles and sometimes in non-right-angled triangles using the sine and cosine rules.

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Core Teaching Sections

Understanding Trigonometric Ratios

The key trigonometric ratios are sine ($\sin$), cosine ($\cos$), and tangent ($\tan$). These ratios help you find unknown sides or angles in right-angled triangles.

• Sine of an angle is the ratio of the opposite side to the hypotenuse: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$.
• Cosine of an angle is the ratio of the adjacent side to the hypotenuse: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$.
• Tangent of an angle is the ratio of the opposite side to the adjacent side: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$.

Applying the Sine and Cosine Rules

For non-right-angled triangles, the sine and cosine rules come in handy:

• Sine Rule: $\frac{𝑎}{\sin 𝐴} = \frac{𝑏}{\sin 𝐵} = \frac{𝑐}{\sin 𝐶}$. Use this when you have two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).
• Cosine Rule: $𝑐^2 = 𝑎^2 + 𝑏^2 - 2ab \cos 𝐶$. Use this when you have two sides and the included angle (SAS) or all three sides (SSS).

Quick Check

Before we dive into examples, try answering these:

1. What is the sine of a 30-degree angle in a right-angled triangle?
2. If $\tan \theta = 1$, what is $\theta$?
3. Can you use the sine rule with two sides and an included angle?

Answers:

1. $\sin 30^\circ = 0.5$
2. $\theta = 45^\circ$
3. No, you need either two angles and one side or two sides and a non-included angle.

Common mistakes students make

One trap is overcomplicating questions that are actually straightforward. Many students rush through algebraic steps, leading to careless errors. Remember, when you see a trigonometric equation, slow down and think about the relationships first.

Another common slip is forgetting to check if your calculator is in the correct mode (degrees instead of radians). This can completely change your answers!

Exam tip

During the exam, allocate time wisely. Spend the first few minutes scanning through all the questions. Start with the ones you find easier to build confidence and momentum. For trigonometry, always sketch a simple triangle if the question allows. This visual can guide you through solving the problem without missing steps.

Worked examples

Question 1

A right-angled triangle has an angle of $30^\circ$ and a hypotenuse of 10 cm. Find the opposite side.

Solution

Step 1: Recognise which trigonometric ratio to use.
Why: The question involves an angle, the hypotenuse, and the opposite side, so we use $\sin$.

Step 2: Write the equation: $\sin 30^\circ = \frac{\text{opposite}}{10}$.
Why: This sets up the relationship between the angle and sides.

Step 3: Solve for the opposite side: $\text{opposite} = 10 \times \sin 30^\circ$.
Why: We rearrange to find the unknown side.

Step 4: Calculate: $\text{opposite} = 10 \times 0.5 = 5$ cm.
Why: Substitute the value of $\sin 30^\circ$ to get the answer.

Question 2

Find the length of side 𝑐 in a triangle where 𝑎 = 7, 𝑏 = 10, and the angle between them is $120^\circ$.

Solution

Step 1: Use the cosine rule: $𝑐^2 = 𝑎^2 + 𝑏^2 - 2ab \cos 𝐶$.
Why: We have two sides and the included angle, so cosine rule fits.

Step 2: Substitute the values: $𝑐^2 = 7^2 + 10^2 - 2 \times 7 \times 10 \times \cos 120^\circ$.
Why: Plug in the given values to find $𝑐^2$.

Step 3: Calculate: $𝑐^2 = 49 + 100 + 140 = 289$.
Why: Simplify the equation step by step.

Step 4: Find 𝑐: $𝑐 = \sqrt{289} = 17$.
Why: Take the square root to solve for 𝑐.

Question 3

In a triangle, $𝐴 = 45^\circ$, $𝐵 = 60^\circ$, and 𝑎 = 5 cm. Find side 𝑏 using the sine rule.

Solution

Step 1: Use the sine rule: $\frac{𝑎}{\sin 𝐴} = \frac{𝑏}{\sin 𝐵}$.
Why: We have two angles and one side, perfect for sine rule.

Step 2: Substitute the known values: $\frac{5}{\sin 45^\circ} = \frac{𝑏}{\sin 60^\circ}$.
Why: Plug in the values to set up the equation.

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Step 3: Solve for 𝑏: $𝑏 = \frac{5 \times \sin 60^\circ}{\sin 45^\circ}$.
Why: Rearrange to find the unknown side.

Step 4: Calculate: $𝑏 = \frac{5 \times \sqrt{3}/2}{\sqrt{2}/2} = \frac{5\sqrt{3}}{\sqrt{2}}$.
Why: Simplify the fraction to get the final answer.

Question 4

Given a triangle with sides 𝑎 = 8, 𝑏 = 15, and 𝑐 = 17, find angle 𝐶 using the cosine rule.

Solution

Step 1: Use the cosine rule to find the angle: $\cos 𝐶 = \frac{𝑎^2 + 𝑏^2 - 𝑐^2}{2ab}$.
Why: We have all sides, so we can find the angle directly.

Step 2: Substitute values: $\cos 𝐶 = \frac{8^2 + 15^2 - 17^2}{2 \times 8 \times 15}$.
Why: Plug in the sides into the formula.

Step 3: Calculate: $\cos 𝐶 = \frac{64 + 225 - 289}{240} = 0$.
Why: Simplify the equation to find $\cos 𝐶$.

Step 4: Find 𝐶: since $\cos 𝐶 = 0$, $𝐶 = 90^\circ$.
Why: An angle with cosine 0 is a right angle.

Quick summary

• Use sine, cosine, and tangent for right-angled triangles.
• Sine and cosine rules help with non-right-angled triangles.
• Always check calculator mode: degrees or radians.
• Draw simple triangles to guide your problem-solving.
• Don’t rush algebra steps; double-check for careless errors.

FAQ

What’s the best way to remember trigonometric ratios?
Think of "SOH-CAH-TOA". It’s a simple mnemonic: Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent.

When do I use the sine rule?
Use it when you know two angles and one side or two sides and a non-included angle.

How do I avoid calculator mistakes?
Always check if your calculator is in degrees mode for trigonometry questions in O Level exams.

Why do I keep losing marks in trigonometry?
Most students lose marks due to rushing steps or not simplifying fractions properly. Practice slowing down and double-checking each step.

Practise with free question sets

Work through more exam-style questions with answers and step-by-step solutions:

• 35+ O Level E Math Trigonometry Questions for 2026/2027 (Singapore MOE Syllabus) with Step-by-Step Solutions
• 35+ O Level AMath Trigonometry Questions for 2026/2027 (Singapore MOE Syllabus) with Full Solutions
• Topic study hub

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Related Topics You Should Learn Next

• O Level A-Math Trigonometry Guide
• Additional Mathematics: Algebra
• O Level A-Math: Differentiation

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