A good right-triangle answer passes three quick tests: the hypotenuse is the longest side, the angles and sides are in the same order, and Pythagoras holds within rounding. Running them takes under a minute and catches most ratio slips.
This lesson closes right-triangle trigonometry by turning checking into a habit you can use after choosing a ratio and finding angles.
What should you check, and why do the checks work?
Every right triangle obeys rules that do not depend on which ratio you used. So they act as an independent test of your answer.
| Check | What to look for |
|---|---|
| Hypotenuse | It is the longest side, opposite the right angle |
| Side and angle order | The longest leg is opposite the larger angle |
| Angle range | Both acute angles are between 0° and 90° and add to 90° |
| Pythagoras | a² + b² is close to c² |
| Ratio range | sin and cos of an acute angle are between 0 and 1 |
| Mode | sin 30 shows 0.5 on your calculator |
How do you check a solution, step by step?
- Write the answer with its unit and rounding.
- Compare it with the hypotenuse and with the other side.
- Estimate: for 45°, the legs are equal; for small angles the opposite side is short.
- Test with Pythagoras if you have found two sides.
- Only then move on, or fix the ratio if a check fails.
Worked example
In a right-angled triangle the angle is 50° and the hypotenuse is 9 cm. A student finds the side opposite the 50° angle and gets 11.75 cm. Is that right?
Check 1, hypotenuse: 11.75 cm is longer than the 9 cm hypotenuse. A leg cannot be longer than the hypotenuse, so the answer fails.
Find the cause: the student wrote 9 ÷ sin 50° = 11.75. The equation sin 50° = x/9 needs a multiplication: x = 9 × sin 50°.
Corrected answer: x = 9 × 0.7660 = 6.894 cm, so 6.89 cm (3 s.f.).
Re-check: the adjacent side is 9 × cos 50° = 9 × 0.6428 = 5.785 cm. Then 6.894² + 5.785² = 47.53 + 33.47 = 81.0, and 9² = 81. ✓ Also 6.89 is larger than 5.79, which fits, since 50° is larger than 40°, so the opposite side is the longer leg.
The mistake to watch for
A common slip is to have the ratio upside down, so the answer breaks the rules. A second slip is checking only that the calculation “felt right”.
Mistaken working: the student finds a leg of 11.75 cm in a triangle with a 9 cm hypotenuse and moves on because the calculator showed a clean number.
A clean number is not evidence. The geometry says the leg must be under 9 cm.
The correction is to make the hypotenuse check automatic. Circle the hypotenuse on the diagram at the start, and compare your answer with it before writing the final line.
Check yourself
1. A right-angled triangle has a hypotenuse of 12 cm and an angle of 70°. A student says the adjacent side is 35.1 cm. Find the error and give the correct answer.
Show answer
35.1 cm is longer than the hypotenuse, so it is impossible. The student divided: 12 ÷ cos 70° = 35.1. The correct working is 12 × cos 70° = 12 × 0.3420 = 4.10 cm.
2. A student finds an angle of 0.775 when using sin⁻¹(0.7) in a triangle question. What has probably gone wrong?
Show answer
The calculator is most likely in radian mode. In degree mode sin⁻¹(0.7) is 44.4°. The value 0.775 is 44.4° written in radians. The test is sin 30, which must give 0.5 in degree mode.
3. Could a right-angled triangle with legs of 5 cm and 7 cm have a hypotenuse of 6 cm? If not, what is the hypotenuse?
Show answer
No, because the hypotenuse must be longer than both legs, including 7 cm. By Pythagoras, c² = 25 + 49 = 74, so c = √74 = 8.60 cm (3 s.f.).
Where this leads next
Put the checks to work in the right-triangle trigonometry practice set. After that, triangles without a right angle come in non-right triangles and bearings. The triangle and bearings reasoning board has plausibility checks of its own, and the non-calculator working trainer supports exact arithmetic checks.
A student who checks well often learned it from watching someone else do it aloud. That is one reason teachers work on it in online one-to-one Mathematics tuition.