The Six Trigonometric Ratios
Simple Explanation
For an acute angle θ in a right triangle, there are six trigonometric ratios, built from the three sides (opposite, adjacent, hypotenuse): sine, cosine, and tangent (SOH-CAH-TOA), plus their three reciprocals — cosecant, secant, and cotangent.
Why Do We Need It?
These six ratios are the core vocabulary of trigonometry — every other trigonometric result in this chapter builds on these six fixed relationships.
See It
A right triangle with angle theta marked at one vertex, and its sides labelled opposite, adjacent, and hypotenuse
Formula
The Six Trigonometric Ratios
sinθ=opp/hyp, cosθ=adj/hyp, tanθ=opp/adj, cscθ=hyp/opp, secθ=hyp/adj, cotθ=adj/opp
For an acute angle θ in a right triangle, each trigonometric ratio is a fixed ratio of two of the triangle's three sides — three basic ratios, and their three reciprocals.
- opp
- — the length of the side opposite angle θ
- adj
- — the length of the side adjacent to angle θ (not the hypotenuse)
- hyp
- — the length of the hypotenuse, opposite the right angle
When to use it: Whenever you know an angle and one side of a right triangle and need another side, or know two sides and need the angle.
Worked Example
Find all six trigonometric ratios
In a right triangle, the side opposite θ is 3, the adjacent side is 4, and the hypotenuse is 5. Find sinθ, cosθ, and tanθ.
Why Does This Work?
Because all right triangles with the same angle θ are similar (AA similarity, from Chapter 8), the ratio of any two sides is always the same fixed number for that θ, no matter how big or small the triangle is — which is exactly what makes each trigonometric ratio a well-defined function of the angle alone.
Real-Life Example
Calculating a ramp's rise from its angle
A wheelchair ramp is built at a fixed angle, and the builder needs to know how much it rises over a given horizontal run.
tanθ = rise/run directly gives the rise, once the angle and the horizontal run are known — a direct real-world use of the tangent ratio.
Practice
In a right triangle, opp=6, adj=8, hyp=10. Find cosθ (as a decimal).
MediumCommon mistake
Mixing up "opposite" and "adjacent" — the opposite side is always across from angle θ, never touching it; the adjacent side touches θ but is not the hypotenuse.
Quick Review
- sinθ=opp/hyp, cosθ=adj/hyp, tanθ=opp/adj (SOH-CAH-TOA).
- cscθ, secθ, cotθ are the reciprocals of sinθ, cosθ, tanθ respectively.
- These ratios are fixed for a given angle θ, regardless of the triangle's size (similar triangles).