Geometry
Sin, Cos & Tan of Triangles
In a right triangle, SOH-CAH-TOA: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. Use one triangle to find all three ratios.
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Worked Examples
Follow along with these step-by-step examples. Take your time — there's no rush!
Problem
A right triangle has legs 3 and 4, hypotenuse 5. Find sin, cos, and tan of the angle opposite the side of length 3.
Step-by-Step Solution
Opposite = 3, Adjacent = 4, Hypotenuse = 5.
sin θ = Opposite/Hypotenuse = 3/5 = 0.6
cos θ = Adjacent/Hypotenuse = 4/5 = 0.8
tan θ = Opposite/Adjacent = 3/4 = 0.75
Answer
sin = 3/5, cos = 4/5, tan = 3/4
Problem
In a right triangle, the angle is 30°, hypotenuse = 10. Find the opposite side.
Step-by-Step Solution
sin 30° = Opposite/Hypotenuse
0.5 = Opposite/10
Opposite = 5
Answer
Opposite side = 5
Problem
In a right triangle, opposite = 7, adjacent = 7. What is the angle?
Step-by-Step Solution
tan θ = 7/7 = 1
θ = tan⁻¹(1) = 45°
Answer
θ = 45°
Problem
A right triangle has hypotenuse 10 and one leg 6. Find the other leg.
Step-by-Step Solution
Use the Pythagorean theorem: a² + b² = c²
6² + b² = 10² → 36 + b² = 100
b² = 64 → b = 8
Answer
Other leg = 8
Problem
In a right triangle, cos θ = 5/13. Find sin θ and tan θ.
Step-by-Step Solution
cos θ = adjacent/hypotenuse = 5/13 → adjacent = 5, hypotenuse = 13.
Use Pythagorean theorem: opposite = √(13² − 5²) = √(169 − 25) = √144 = 12.
sin θ = 12/13, tan θ = 12/5.
Answer
sin θ = 12/13, tan θ = 12/5
Your Turn — Practice Problems
Try all 5 problems on your own first. Write out your work — that's how it sticks!
💡 Tip: Don't peek at the answers until you've genuinely tried each one.
Right triangle: opposite = 5, hypotenuse = 13. Find sin θ.
Right triangle: adjacent = 12, hypotenuse = 13. Find cos θ.
Right triangle: opposite = 5, adjacent = 12. Find tan θ.
sin 45° = ? (exact)
cos 60° = ?
tan 45° = ?
sin 30° = ?
cos 30° = ?
Right triangle: opposite = 8, hypotenuse = 10. Find sin θ.
Right triangle: adjacent = 7, hypotenuse = 25. Find cos θ.
Right triangle: legs 6 and 8. Find the hypotenuse.
Right triangle: opposite = 3, adjacent = 4. Find tan θ.
Finished all 12? Give yourself a pat on the back — then check your work!
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