Sine, Cosine & Tangent

The ratios depend on the angle alone, not on the size. Scale the triangle up and every side grows by the same factor, so the ratios between them are unchanged. That is why a table of sines works for any triangle with that angle, large or small.

SOH CAH TOA names which two sides each ratio uses. Opposite is across from the angle, adjacent is beside it, and the hypotenuse is always the longest side, opposite the right angle. Identify the three sides first and the rest is arithmetic.

Sine and cosine can never exceed 1, but tangent has no limit. The opposite and adjacent sides are always shorter than the hypotenuse, so those two ratios are capped. Tangent divides one short side by the other, and as θ approaches 90° the adjacent side shrinks toward zero and the tangent runs away.

• Try this: Set θ to 45°. Which two ratios become equal, and what is the third?
• Try this: Change the hypotenuse without touching the angle. Which readouts move and which stay put?
• Try this: Push θ toward 85°. What is happening to tan θ, and why?