Right-triangle trigonometry
Choose a trigonometric ratio by identifying the sides relative to an angle.
Before you begin
- Right triangles
- Ratios and calculator use
Name the sides relative to the angle
For a chosen acute angle in a right triangle, the opposite side lies across from it, the adjacent side touches it, and the hypotenuse lies opposite the right angle. Opposite and adjacent swap when you choose the other acute angle. The hypotenuse does not change. Draw and label the triangle before selecting a ratio.
Select and rearrange the right ratio
Use sine for opposite/hypotenuse, cosine for adjacent/hypotenuse and tangent for opposite/adjacent. Choose a ratio containing the given side and the unknown side. Use degree mode for angles given in degrees. If finding an angle, use the appropriate inverse function rather than taking a reciprocal.
Worked example
A 10 m ladder makes a 30° angle with level ground. How high does it reach on a vertical wall?
Show the worked solution
- The ladder is the hypotenuse and the height is opposite the 30° angle.
- Write sin 30° = h/10.
- Multiply by 10: h = 10 × 0.5.
5 m.
Try it yourself
Which ratio uses only opposite and adjacent?
Common mistakes
- SOH–CAH–TOA applies directly to right triangles only.
What can you explain now?
A ramp rises 3 m over a horizontal distance of 4 m. Find its angle to the horizontal to one decimal place.
Compare with the explanation
36.9°.
tan θ = 3/4, so θ = arctan(0.75) ≈ 36.8699°. The horizontal distance is adjacent, not the ramp length.
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Original StudyForA content. AI-assisted checks completed; subject-teacher review is still pending. These lessons are not endorsed by an exam board.
Common Core mathematics standards ↗