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Right-triangle trigonometry practice
Sine, cosine and tangent compare side lengths relative to a chosen acute angle in a right triangle. Label the sides after choosing the angle, not before.
By ExamCat · Original study lesson · Published September 24, 2026
Understand the method
The hypotenuse is opposite the right angle. For an acute angle θ, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent. The two legs exchange opposite and adjacent roles when the reference angle changes.
Worked example, step by step
A right triangle has opposite side 3, adjacent side 4 and hypotenuse 5 relative to angle θ.
- Calculate sin θ = 3/5 = 0.6.
- Calculate cos θ = 4/5 = 0.8.
- Calculate tan θ = 3/4 = 0.75.
- Using inverse tangent in degree mode gives θ ≈ 36.9°. The other acute angle is about 53.1°.
A mistake to watch for
A calculator in radians will report a different number for the same angle. Also, inverse sine is not the reciprocal of sine: arcsin finds an angle, whereas 1/sin is a reciprocal ratio.
Try it yourself: three questions
Write an answer before opening the explanation. For calculations, keep your units and intermediate steps; for concepts, state the reason for your choice.
1. Hypotenuse is 10 and an acute angle is 30°. Find the opposite side.
Show answer and explanation
10 sin 30° = 5. The hypotenuse belongs in the denominator of the initial ratio.
2. Opposite is 5 and adjacent is 12. Find tan θ.
Show answer and explanation
5/12. You do not need the hypotenuse to calculate tangent.
3. Can an acute angle in a right triangle have sin θ = 1.2?
Show answer and explanation
No. The opposite leg is shorter than the hypotenuse, so this ratio must lie between zero and one.
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Mark the first step where your reasoning differed from the explanation. Turn that step into a question, answer it without looking, then change one condition in the example and explain what would change. A correct guess is a reason to revisit the method.
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