A reliable method
Mark the right angle and the chosen acute angle. Identify opposite, adjacent and hypotenuse relative to that angle before selecting a ratio.
The common trap
The hypotenuse is opposite the right angle and is the longest side. Changing the chosen angle swaps the opposite and adjacent legs.
Make the connection
Check calculator degree mode for degree questions. Exact special triangles offer independent checks for familiar angles.
Select the correct side relationship
A right triangle with legs six and eight has hypotenuse ten because 6²+8²=100. Adding the leg lengths would give fourteen, which does not satisfy Pythagoras. If the named angle has adjacent leg six, its cosine is 6/10=3/5; its sine is 8/10=4/5. Both ratios are valid probabilities-sized numbers, but they answer different trigonometric questions. A missing leg is found by subtracting its partner’s square from the hypotenuse square. Draw and label the right angle before choosing addition or subtraction.
Transfer, explain and check
A triangle has hypotenuse 13 and one leg 5. The other leg is √(169−25)=12. For the angle adjacent to the side of length 5, cosine is 5/13. Explain why 12/13 is sine for that same angle.
A right triangle has perpendicular sides 9 cm and 12 cm. Find its hypotenuse length in cm.
- Square and add the perpendicular side lengths.
- Take the positive square root.
- Check that the result exceeds both given sides.
Compare the other answer choices
Subtract the two leg lengths.
Add leg lengths instead of combining their squares.
Stop at the squared hypotenuse.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Cambridge IGCSE Mathematics 0580