Trigonometry in Right Triangles

Aug 19, 2025

Overview

This lecture demonstrates the practical use of trigonometric functions to solve for unknown sides in right triangles using given angles and side lengths.

Example 1: Using Cosine to Find an Adjacent Side

  • Given a right triangle with angle π/4 radians (45°) and hypotenuse 10√2.
  • The goal is to find the length of the adjacent side to the given angle.
  • Cosine relates the adjacent side to the hypotenuse: cos(θ) = adjacent/hypotenuse.
  • SOHCAHTOA mnemonic helps identify cosine for adjacent and hypotenuse (CAH).
  • Set up the equation: cos(π/4) = a / (10√2).
  • Rearranged: a = 10√2 × cos(π/4).
  • Cos(π/4) = √2/2.
  • Multiply: a = 10√2 × (√2/2) = 10 (since √2 × √2 = 2 and 2/2 = 1).
  • Therefore, the adjacent side is 10 units long.

Example 2: Using Tangent to Find an Adjacent Side

  • Given a right triangle with angle 0.54 radians and opposite side of length 3.
  • The goal is to find the length of the adjacent side.
  • Tangent relates the opposite and adjacent sides: tan(θ) = opposite/adjacent.
  • SOHCAHTOA mnemonic identifies tangent (TOA) for opposite and adjacent.
  • Set up the equation: tan(0.54) = 3 / a.
  • Rearranged: a = 3 / tan(0.54).
  • Provided tan(0.54) = 3/5.
  • Calculate: a = 3 / (3/5) = 3 × (5/3) = 5.
  • Therefore, the adjacent side is 5 units long.

Problem Solving Steps with Trig Functions

  • Identify which sides (opposite, adjacent, hypotenuse) are known or unknown.
  • Use SOHCAHTOA to select the correct trigonometric function.
  • Write the equation involving the angle and sides.
  • Rearrange to solve for the unknown side.
  • Substitute known values and calculate or use a calculator for trig values.

Key Terms & Definitions

  • Radians — A unit for measuring angles; π radians = 180 degrees.
  • Hypotenuse — The longest side of a right triangle, opposite the right angle.
  • Adjacent Side — The side next to the given angle (except the hypotenuse).
  • Opposite Side — The side across from the given angle.
  • SOHCAHTOA — Mnemonic for sine, cosine, and tangent ratios.
  • Cosine (cos) — Ratio of adjacent side to hypotenuse.
  • Tangent (tan) — Ratio of opposite side to adjacent side.

Action Items / Next Steps

  • Practice more triangle problems using trig functions as shown.
  • Remember to use a calculator set to radians when working with angle measures in radians.

Full transcript