📐

Right Triangle Solving Techniques

Sep 3, 2025

Overview

This lecture explains how to solve right triangles when given partial information, using angle sums, trigonometric ratios (SOHCAHTOA), and the Pythagorean theorem.

Angle Sums in Right Triangles

  • The angles in any triangle add up to 180 degrees.
  • In a right triangle, one angle is always 90 degrees.
  • The sum of the two remaining angles is 90 degrees (i.e., each must be less than 90).

Labeling Sides and Angles

  • Each angle is labeled (example: A, B, C), with angle C typically as the right angle (90°).
  • Side labels are lowercase and opposite their respective angles (side c is opposite angle C, the hypotenuse).

Using SOHCAHTOA

  • Sine (sin) relates opposite side and hypotenuse: sin(angle) = opposite/hypotenuse.
  • Cosine (cos) relates adjacent side and hypotenuse: cos(angle) = adjacent/hypotenuse.
  • Tangent (tan) relates opposite and adjacent sides: tan(angle) = opposite/adjacent.

Example Problem 1 – Solving for Unknown Sides and Angles

  • Given: One side length (b = 12) and one angle (other than the right angle, e.g., 30°).
  • Find missing angle: Subtract known angles from 180° to get the third angle.
  • Label triangle and sides.
  • Use cosine to find the hypotenuse: cos(30°) = adjacent/hypotenuse ⇒ hypotenuse = 12 / cos(30°).
  • Use Pythagorean theorem: a² + b² = c² to find missing side.
  • Simplify square roots as needed for final answers.

Example Problem 2 – Given Two Sides, Find Angles

  • Given: side a = 21, hypotenuse c = 29.
  • Use Pythagorean theorem to find the missing side: b² = c² – a².
  • Use tangent to find missing angle: tan(A) = opposite/adjacent = 21/20 ⇒ A = arctan(21/20).
  • Find the third angle: 90° – known acute angle.

Key Terms & Definitions

  • Right Triangle — a triangle with one 90° angle.
  • Hypotenuse — the side opposite the right angle; the triangle’s longest side.
  • Pythagorean theorem — states a² + b² = c² for right triangles.
  • SOHCAHTOA — mnemonic for sine, cosine, and tangent trigonometric ratios.
  • Inverse trigonometric function — finds angles from known side ratios (e.g., arctan, arcsin, arccos).

Action Items / Next Steps

  • Practice solving right triangles using given side lengths and/or angles.
  • Ensure calculator is in degree mode when solving for angles.
  • Review SOHCAHTOA and the Pythagorean theorem for additional exercises.

Full transcript