Solving Right Triangles
Simple Explanation
"Solving" a right triangle means finding every missing side and angle. Given enough information (one side plus one other side or angle), you can find everything else using the trigonometric ratios, the Pythagorean theorem, and the fact that the two acute angles sum to 90°.
Why Do We Need It?
This combines everything from this chapter into one complete practical skill — the core technique behind surveying, navigation, and construction measurements.
See It
A right triangle with a 35-degree angle marked and the hypotenuse labelled 10, with the other two sides unlabelled to be found
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
Solve a right triangle
In right triangle ABC (right angle at B), ∠C = 35° and hypotenuse AC = 10. Find AB, BC, and ∠A.
Why Does This Work?
A right triangle has only 3 independent pieces of information (out of its 3 angles and 3 sides, since one angle is always 90° and the other two always sum to 90°) — once one side and one other measurement are known, the trigonometric ratios and the angle sum rule are always enough to solve for everything else.
Real-Life Example
Surveying an inaccessible distance
A surveyor measures one angle and one distance to a landmark across a river, without being able to physically cross it.
Solving the resulting right triangle using trigonometric ratios gives every other distance and angle needed, entirely from the surveyor's side of the river.
Practice
In a right triangle, one acute angle is 40° and the adjacent side is 12. Find the opposite side. (tan40° ≈ 0.839. Round to 1 decimal place.)
HardCommon mistake
Choosing the wrong trigonometric ratio for the given information — always identify which sides are known/unknown relative to the given angle (opposite, adjacent, or hypotenuse) before picking sin, cos, or tan.
Quick Review
- Use sin, cos, or tan to find a missing side from a known angle and one known side.
- The two acute angles of a right triangle always sum to 90°.
- Use the Pythagorean theorem as a check, or to find a third side once two are known.