Rearrange a trigonometric ratio to find a side
Choose a ratio with one unknown side · Rearrange before using a calculator · Round only the final requested quantity
Section C8.1 · Core · Right triangles, bearings, elevation and depression
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
- Trigonometric side roles
- Equation rearrangement
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
- Choose a ratio with one unknown side
- Rearrange before using a calculator
- Round only the final requested quantity
Write an equation before entering numbers
If an acute angle θ and adjacent leg a are given, opposite o=a tan θ and hypotenuse h=a/cos θ. If opposite is known, h=o/sin θ and a=o/tan θ. Multiplying instead of dividing in the hypotenuse equation can produce a length smaller than a leg.
Degree mode and precision
Use degree mode for these numerical tasks when calculator use is allowed. Keep the unrounded trigonometric value and intermediate lengths until the final result; the examples request three significant figures. If a calculator is not allowed, keep the required exact expression or use information supplied by that question. These exercises are original topic practice, not a timed non-calculator paper.
Sources for this lesson
Pause and explain
From cos θ=a/h, how do you obtain h?
Worked example
A right triangle has θ=35° and adjacent leg 10 cm. Find opposite and hypotenuse to 3 significant figures.
Show the worked solution
- Opposite=10 tan35°≈7.002075 cm.
- Hypotenuse=10/cos35°≈12.207746 cm.
Answer Opposite 7.00 cm; hypotenuse 12.2 cm.
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.
A right triangle has angle 45° and adjacent leg 8 cm. Find opposite and hypotenuse to 3 significant figures.
Give me a hint
Use tangent for the opposite and cosine for the hypotenuse.
Compare my reasoning
- Opposite=8 tan45°=8 cm.
- Hypotenuse=8/cos45°≈11.313708 cm.
Opposite 8.00 cm; hypotenuse 11.3 cm.
Look for these in your work
- Rearrange the chosen ratio
- Round the final lengths
A right triangle has angle 40° and opposite leg 6 m. Find adjacent and hypotenuse to 3 significant figures.
Give me a hint
Both unknowns require division by the appropriate ratio.
Compare my reasoning
- Adjacent=6/tan40°≈7.150522 m.
- Hypotenuse=6/sin40°≈9.334343 m.
Adjacent 7.15 m; hypotenuse 9.33 m.
Look for these in your work
- Select the two correct ratios
- Check the hypotenuse is longest
A 5 m ladder forms a 65° angle with level ground and meets a vertical wall. Find vertical reach and horizontal distance to 3 significant figures.
Give me a hint
The ladder is the hypotenuse.
Compare my reasoning
- Vertical reach=5 sin65°≈4.531539 m.
- Horizontal distance=5 cos65°≈2.113091 m.
Vertical reach 4.53 m; horizontal distance 2.11 m.
Look for these in your work
- Identify the given angle and hypotenuse
- State the ideal right-angle model
Common mistakes
- Use degrees for degree inputs.
- A hypotenuse should exceed the known leg.
What can you explain now?
From cos θ=a/h, how do you obtain h?
Compare with the explanation
h=a/cos θ.
Multiply by h and then divide by the positive acute-angle cosine.
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Make it stick
From cos θ=a/h, how do you obtain h?
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.