Algebra 2 · Unit 10 · Topic 10.3
Right-Triangle Ratios and Tangent
For an acute angle in a right triangle, sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent.
Definition and explanation
For an acute angle in a right triangle, sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. More generally tan θ = sin θ/cos θ wherever cos θ ≠ 0.
Worked example
A right triangle has legs 5 and 12 and hypotenuse 13. For the angle opposite the side of length 5, sin θ = 5/13, cos θ = 12/13, and tan θ = 5/12. Their values depend on which acute angle is chosen.
Common mistake
Opposite and adjacent are defined relative to the selected angle; the hypotenuse is always opposite the right angle.
Key takeaway
For an acute angle in a right triangle, sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.