Right Triangle Trigonometry Overview

Overview

This lecture covers right triangle trigonometry, focusing on the six trigonometric functions, how to apply SOHCAHTOA, special right triangle ratios, and how to find missing sides and angles in right triangles.

Right Triangle Basics

• Right triangles have one 90° angle and three sides: opposite, adjacent, and hypotenuse (longest side).
• The Pythagorean theorem states: (a^2 + b^2 = c^2) for side lengths (a, b), and hypotenuse (c).

The Six Trigonometric Functions

• SOHCAHTOA helps recall the basic trig ratios:
  • Sine (sin θ) = opposite / hypotenuse
  • Cosine (cos θ) = adjacent / hypotenuse
  • Tangent (tan θ) = opposite / adjacent
• The reciprocal functions:
  • Cosecant (csc θ) = 1 / sin θ = hypotenuse / opposite
  • Secant (sec θ) = 1 / cos θ = hypotenuse / adjacent
  • Cotangent (cot θ) = 1 / tan θ = adjacent / opposite

Special Right Triangle Triplets

• Common Pythagorean triplets: 3-4-5, 5-12-13, 7-24-25, 8-15-17.
• Multiples of these triplets also form right triangles.

Example Problems: Finding Sides and Trigonometric Values

• Given two sides, use the Pythagorean theorem to find the third.
• Calculate all six trig functions using side lengths and SOHCAHTOA.
• Reduce fractions when possible for simplified trig ratios.

Finding Sides with Trig Functions

• Use trig functions to solve for missing sides:
  • For opposite and adjacent, use tangent.
  • For adjacent and hypotenuse, use cosine.
  • For opposite and hypotenuse, use sine.
• Use a calculator for decimal answers, ensure degree mode is set.

Finding Angles with Trig Functions

• Use the inverse trig functions (arcsin, arccos, arctan) to find angles when given sides.
• Enter the ratio into the calculator using the appropriate inverse function for the desired angle.

Key Terms & Definitions

• Opposite side — side across from the angle of interest.
• Adjacent side — side next to the angle of interest (not the hypotenuse).
• Hypotenuse — the longest side in a right triangle, across from the right angle.
• SOHCAHTOA — mnemonic for remembering trig ratios: Sine, Cosine, Tangent.
• Reciprocal functions — cosecant, secant, cotangent (the flips of sine, cosine, tangent).

Action Items / Next Steps

• Practice finding missing sides and angles using trig functions and the Pythagorean theorem.
• Memorize SOHCAHTOA and common Pythagorean triplets.
• Use a calculator in degree mode for non-whole number answers.
• Review assigned readings or problem sets related to right triangle trigonometry.