Sine, cosine and tangent for missing sides
Label sides relative to the chosen acute angle and choose the ratio containing the known and unknown sides.
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Identify a hypotenuse.
- Rearrange equations and use a calculator in degrees.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which ratio uses opposite and hypotenuse?
What you will learn
- Identify opposite, adjacent and hypotenuse.
- Use trigonometric ratios to find a right-triangle side.
The chosen angle decides opposite and adjacent
The hypotenuse H is fixed by the right angle. Relative to a chosen acute angle θ, O is the side opposite it and A is the neighbouring leg. The neighbouring hypotenuse does not count as the adjacent leg. Changing the chosen acute angle swaps O and A while H stays fixed.
The ratios are sin θ = O/H, cos θ = A/H and tan θ = O/A. Choose the one containing the two relevant sides. If θ = 30° and H = 10 cm, O = 10 sin 30° = 5 cm. If θ = 40° and A = 7 cm, O = 7 tan 40° ≈ 5.87 cm. Calculator mode must be degrees.
Rearrange before pressing the keys
An unknown numerator usually comes from multiplication: A = H cos θ. An unknown denominator requires division: H = O/sin θ or A = O/tan θ. For O = 6 and θ = 30°, H = 6/0.5 = 12 cm. Multiplying would incorrectly give three, shorter than the given leg.
Draw a side-labelled right triangle, write one ratio equation and isolate the unknown. A quick size check helps: a sine or cosine ratio for an acute angle is between zero and one, so each leg is shorter than H. Retain calculator precision during intermediate steps and round only the final answer.
Pause and explain
Angle θ = 30° and hypotenuse = 10 cm. Find the opposite side.
Worked example
In a right triangle, θ = 30° and the opposite side is 6 cm. Find the hypotenuse.
Show the worked solution
- The relevant ratio is sin θ = O/H.
- Rearrange H = O/sin θ = 6/sin 30° = 12.
- Twelve exceeds the given leg six, consistent with a hypotenuse.
Answer 12 cm.
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Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Angle θ = 60° and hypotenuse = 14 cm. Find the adjacent side.
Give me a hint
Cosine uses adjacent over hypotenuse.
Compare my reasoning
- cos 60° = A/14.
- A = 14 × 0.5 = 7 cm.
- Seven is shorter than the hypotenuse fourteen.
7 cm.
Look for these in your work
- Labelled the adjacent leg correctly.
- Multiplied the known hypotenuse by cosine.
Angle θ = 45° and adjacent leg = 9 cm. Find the opposite leg.
Give me a hint
Tangent relates the two legs.
Compare my reasoning
- tan 45° = O/9.
- Since tan 45° = 1, O = 9 cm.
- The two equal legs agree with the 45° right-triangle shape.
9 cm.
Look for these in your work
- Used tangent rather than a hypotenuse ratio.
- Checked the equal-leg interpretation.
A 10 m ramp rises at 30° to horizontal. Find vertical rise and horizontal run.
Give me a hint
The ramp itself is the hypotenuse.
Compare my reasoning
- Vertical rise is 10 sin 30° = 5 m.
- Horizontal run is 10 cos 30° = 5√3 m ≈ 8.66 m.
- Check 5² + (5√3)² = 100 for the 10 m hypotenuse.
Rise 5 m; run 5√3 m ≈ 8.66 m.
Look for these in your work
- Distinguished ramp length from horizontal run.
- Used sine and cosine for the appropriate legs.
Common mistakes
- Counting the hypotenuse as adjacent.
- Using the wrong reference angle.
- Multiplying when the unknown is a denominator.
What can you explain now?
What calculator angle mode is needed for these syllabus angles?
Compare with the explanation
Degrees.
The supplied and requested angles are measured in degrees; radian mode gives different ratio values.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, label O, A and H for both acute angles of one triangle, then explain which labels change.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.