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Sine, Cosine, Tangent

Overview

SOH CAH TOA.

What is it?

The three primary ratios of side lengths in a right-angled triangle.

History

Ancient Indians and Arabs refined these ratios for astronomy and navigation long before calculators.

Key Idea

SOH (Sin=Opp/Hyp), CAH (Cos=Adj/Hyp), TOA (Tan=Opp/Adj).

Practice This Topic

Concept Guide

Plain English: Triangles have a secret code. If you know one angle, the ratio of the sides is locked in stone. 'Sine' is just a fancy word for how big the opposite side is compared to the slant (hypotenuse).

Real-world example: Carpentry. Calculating the exact angle to cut a rafter so the roof fits perfectly.

How to do it

  1. Identify your 'Target Angle' (theta).
  2. Label the three sides: Hypotenuse (longest), Opposite (across from angle), Adjacent (touching angle).
  3. Pick the function that matches the two sides you know or need (use SOH CAH TOA).
  4. Write the equation and solve for x.
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Common Pitfall

Calculator mode! Using Degrees mode for Radian problems (or vice versa) will give you the wrong answer every time.

Word Problem
"You are placing a 12-foot ladder against a wall. Safety rules say the bottom angle must be 70 degrees. How high up the wall will the ladder reach?"
Reasoning: We know Hypotenuse (12) and Angle (70). We want Opposite (height). SOH says Sin(70) = x/12. So, x = 12 * Sin(70). The ladder reaches approx 11.28 feet.

Practice Examples

The Setup
/| H / | / | O /__ | A
H=Hypotenuse, O=Opposite, A=Adjacent
SOH Formula
Opposite Sin = ---------- Hypotenuse
Use when you don't care about the Adjacent side.