Inverse ratios for right-triangle angles
Use inverse sine, cosine or tangent and round an angle in degrees to one decimal place.
Section Paper 4 · Extended · Calculator
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
- Label opposite, adjacent and hypotenuse.
- Use degree mode and inverse calculator functions.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which inverse function uses opposite divided by adjacent?
What you will learn
- Choose an inverse ratio from two known sides.
- Check an acute result and avoid treating inverse sine as a reciprocal.
Recover the angle from a ratio
If opposite and adjacent legs are 3 and 4 cm, tan θ = 3/4 and θ = tan⁻¹(3/4) ≈ 36.9°. If opposite and hypotenuse are known, use sin⁻¹(O/H); for adjacent and hypotenuse use cos⁻¹(A/H). Write the ratio in the chosen angle's order before applying the inverse function.
The inverse notation sin⁻¹ here means the angle whose sine is the supplied ratio, not 1/sin. For a right triangle's acute angle, sine and cosine inputs must lie strictly between zero and one. A ratio greater than one suggests a swapped hypotenuse or incompatible measurements.
Check the geometry and accuracy
The two acute angles sum to 90°. For a 3–4–5 triangle, the angle opposite side three is about 36.9° and the angle opposite four is about 53.1°. A larger side faces the larger angle, giving another useful check without repeating every calculation.
Use full calculator precision for the ratio and intermediate angle. Give final decimal angles in degrees to one decimal place unless a question specifies otherwise. A right-angle mark is essential to these ratios; two arbitrary sides of a non-right triangle need the sine or cosine rule later in Extended.
Pause and explain
A right triangle has O = 3 and H = 6. Find the chosen acute angle.
Worked example
A right triangle has opposite leg 3 cm and adjacent leg 4 cm relative to θ. Find θ.
Show the worked solution
- tan θ = 3/4 = 0.75.
- Use inverse tangent in degrees: θ ≈ 36.8699°.
- Round the final angle to one decimal place: 36.9°.
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.
A right triangle has A = 4 and H = 8. Find θ.
Give me a hint
Use the adjacent-to-hypotenuse ratio.
Compare my reasoning
- cos θ = 4/8 = 0.5.
- θ = cos⁻¹(0.5) = 60°.
- The other acute angle is 30°, confirming the right-triangle total.
60°.
Look for these in your work
- Selected cosine from the known sides.
- Used an inverse function for an angle.
A right triangle has O = 8 and A = 15. Find θ to one decimal place.
Give me a hint
Apply inverse tangent to 8/15.
Compare my reasoning
- tan θ = 8/15.
- Inverse tangent gives approximately 28.0725°.
- The final angle is 28.1°, smaller than 45° because O is shorter than A.
28.1°.
Look for these in your work
- Kept the ratio in the correct order.
- Rounded the final degree angle only.
A ramp rises 1.5 m over a horizontal distance of 4 m. Find its angle to horizontal.
Give me a hint
Horizontal run is adjacent, not hypotenuse.
Compare my reasoning
- tan θ = 1.5/4.
- θ = tan⁻¹(0.375) ≈ 20.5560°.
- To one decimal place the slope angle is 20.6°.
20.6°.
Look for these in your work
- Used run and rise as perpendicular legs.
- Reported the angle in degrees.
Common mistakes
- Reporting the ratio as an angle.
- Using inverse sine for two known legs.
- Rounding the ratio early or using radian mode.
What can you explain now?
What does sin⁻¹(0.5) mean in an angle calculation?
Compare with the explanation
The angle whose sine is 0.5.
The inverse function reverses sine on the appropriate angle range; it is not a reciprocal.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Tomorrow, choose two sides of a right triangle and recover each acute angle independently.
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.