Choose sine, cosine and tangent relative to the named angle
Identify opposite, adjacent and hypotenuse · Select the ratio that uses the supplied sides · Reassign opposite and adjacent for the other acute angle
Section Paper 2 · Historical Extended · non-calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Right-triangle geometry
- Fractions
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Identify opposite, adjacent and hypotenuse
- Select the ratio that uses the supplied sides
- Reassign opposite and adjacent for the other acute angle
Name the sides relative to θ
In a non-degenerate right triangle, sin θ=opposite/hypotenuse, cos θ=adjacent/hypotenuse and tan θ=opposite/adjacent. The adjacent leg is next to θ but is not the hypotenuse. For the illustrated θ, the opposite leg is 5, adjacent leg 12 and hypotenuse 13.
The hypotenuse stays the same for either acute angle. Opposite and adjacent exchange when the other acute angle is selected. A side is not permanently called opposite without naming the angle.
Match the available information
Use sine for opposite and hypotenuse, cosine for adjacent and hypotenuse, and tangent for the two legs. For an acute angle, sine and cosine are between 0 and 1, while tangent can exceed 1. If a proposed hypotenuse is shorter than a leg, the right-triangle data are inconsistent.
Sources for this lesson
Pause and explain
Does the opposite leg stay the same when you change to the other acute angle?
Worked example
For the illustrated θ, write the three ratios.
Show the worked solution
- Opposite=5, adjacent=12, hypotenuse=13.
- Form the fractions in the stated roles.
Answer sin θ=5/13; cos θ=12/13; tan θ=5/12.
Try a topic test
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.
Relative to θ, a right triangle has opposite 3, adjacent 4 and hypotenuse 5. Write its three ratios.
Give me a hint
Use the definitions directly.
Compare my reasoning
- sin θ=3/5; cos θ=4/5.
- tan θ=3/4.
sin θ=3/5; cos θ=4/5; tan θ=3/4.
Look for these in your work
- Match each side role
- Use the correct denominator
For the other acute angle in the illustrated 5–12–13 triangle, write sine, cosine and tangent.
Give me a hint
Exchange the two leg roles.
Compare my reasoning
- The opposite is now 12 and adjacent 5.
- Hypotenuse remains 13.
Sine 12/13; cosine 5/13; tangent 12/5.
Look for these in your work
- Reassign the angle-dependent roles
- Keep the hypotenuse unchanged
A student uses opposite 12 and hypotenuse 10 and obtains sine 1.2 for an acute right-triangle angle. Can these data describe that triangle?
Give me a hint
Compare the proposed leg with the hypotenuse.
Compare my reasoning
- The hypotenuse must be longer than either leg.
- A ratio 1.2 also exceeds the sine range, so the data are inconsistent.
No; the proposed hypotenuse is too short and the sine ratio is impossible.
Look for these in your work
- Check the geometry before calculating
- Use the ratio range
Common mistakes
- Adjacent does not mean the hypotenuse.
- Opposite depends on the selected angle.
What can you explain now?
Does the opposite leg stay the same when you change to the other acute angle?
Compare with the explanation
No; the two legs exchange opposite and adjacent roles.
Each role is defined relative to the angle being considered.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Does the opposite leg stay the same when you change to the other acute angle?
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.