Recipe

Solve a right triangle

Use when: you're given a right triangle with two known pieces of information beyond the free 90° angle (two sides, or one side and one acute angle) and need every side length and angle measure

Goal: all three side lengths and both acute angle measures

Taught in Solving right triangles

1. 1

   Write down exactly which two pieces of information the problem gives you beyond the right angle: either two sides, or one side and one acute angle. That choice decides every tool used from here on.

   • two sides are known — find the third side with the Pythagorean theorem (Step 2), then find an acute angle with inverse trig (Step 3).
   • one side and one acute angle are known — find each remaining side with a trig ratio (Step 2), then get the second acute angle for free (Step 4).

   Why: guessing a ratio before identifying what's known is the single most common source of wrong answers in this chapter, since the two starting cases call for different tools.

2. 2

   If two sides are known, get the third from the Pythagorean theorem, a^2+b^2=c^2. If one side and one acute angle are known, pick the ratio (sine, cosine, or tangent) that connects the known side to the unknown side, then solve the resulting equation.

   Example

   Hypotenuse 22, angle 48°, opposite leg unknown: sine connects them, so \text{opposite} = 22\sin 48° \approx 16.35.

   Why: the Pythagorean theorem never needs an angle, so it's the faster tool whenever two sides are already available. A trig ratio is required the moment only one side is known, since finding the other side needs a known angle to work with.

3. 3

   Once at least two sides are known (handed to you, or just found in Step 2), pick the ratio those two sides form, then apply its inverse: \sin^{-1}, \cos^{-1}, or \tan^{-1}.

   Example

   Legs 9 and 12 known, angle \theta opposite the 12 unknown: \tan\theta = \tfrac{12}{9}, so \theta = \tan^{-1}\!\left(\tfrac{12}{9}\right) \approx 53.1°.

   Why: an inverse trig ratio is exactly the tool built to reverse "angle in, ratio out" back into "ratio in, angle out."

4. 4

   Subtract the known (or just-found) acute angle from 90°.

   Example

   One acute angle is 48°, so the other is 90° - 48° = 42°, no further computation needed.

   Why: a triangle's three angles always sum to 180°, and 90° of that is already spent on the right angle, leaving exactly 90° split between the other two.