Use inverse ratios and check the angle units and range
Use inverse sine, cosine or tangent for an unknown acute angle · Check whether the supplied ratio is geometrically possible · Distinguish degree results from radian results
Section Paper 3 · Historical Core · 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
- Trigonometric ratios
- Right-triangle angle sum
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
- Use inverse sine, cosine or tangent for an unknown acute angle
- Check whether the supplied ratio is geometrically possible
- Distinguish degree results from radian results
Undo the ratio
For an unknown acute θ, θ=sin⁻¹(o/h), cos⁻¹(a/h), or tan⁻¹(o/a), using degree mode for degree answers. The inverse symbol undoes a function; it does not mean 1/sin θ. For opposite 3 and adjacent 4, tan⁻¹(3/4)≈36.8699°.
Check before accepting the display
An acute right-triangle sine or cosine ratio must lie strictly between 0 and 1. A proposed leg larger than the hypotenuse makes the geometry impossible. A result of about 0.644 radians for tan⁻¹(3/4) is consistent with 36.9°, but 0.644° is a different angle. The other acute angle is 90°−θ.
Sources for this lesson
Pause and explain
Can sin⁻¹(1.2) give an acute real right-triangle angle?
Worked example
Find the acute angle with opposite 3 cm and adjacent 4 cm, to 1 decimal place.
Show the worked solution
- tan θ=3/4.
- Use inverse tangent in degree mode.
Answer 36.9°.
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.
Find the acute angle with opposite 9 cm and adjacent 12 cm, to 1 decimal place.
Give me a hint
Use inverse tangent of 9/12.
Compare my reasoning
- tan θ=0.75.
- θ≈36.8699° rounds to 36.9°.
36.9°.
Look for these in your work
- Use the matching inverse
- Report degree units
A right triangle has opposite 8 cm and hypotenuse 17 cm relative to θ. Find θ and the other acute angle to 1 decimal place.
Give me a hint
Use inverse sine, then subtract from 90°.
Compare my reasoning
- θ=sin⁻¹(8/17)≈28.0725°.
- The other angle is approximately 61.9275°.
28.1° and 61.9°.
Look for these in your work
- Use the valid ratio
- Keep precision before subtraction
A calculator reports 0.643501 radians for tan⁻¹(3/4). A student writes 0.644°. Correct the result to 1 decimal place.
Give me a hint
Convert radians by multiplying by 180/π.
Compare my reasoning
- 0.643501×180/π≈36.8699°.
- The rounded degree result is 36.9°, not 0.644°.
36.9°.
Look for these in your work
- Distinguish angle units
- Do not relabel radians as degrees
Common mistakes
- Inverse sine is not a reciprocal.
- Check calculator mode before reporting an angle.
What can you explain now?
Can sin⁻¹(1.2) give an acute real right-triangle angle?
Compare with the explanation
No; 1.2 is outside the real sine range.
A leg-to-hypotenuse ratio cannot exceed 1.
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Make it stick
Can sin⁻¹(1.2) give an acute real right-triangle 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.