Right Triangle Trigonometry Overview

Overview

This lecture covers right triangle trigonometry, focusing on the six trigonometric functions, how to use them to find missing sides and angles, and special right triangle relationships.

Right Triangle Basics

• A right triangle has three sides: opposite, adjacent, and hypotenuse.
• The hypotenuse is the longest side, always opposite the right angle.
• The Pythagorean Theorem: a² + b² = c², where c is the hypotenuse.

Trigonometric Functions (SOHCAHTOA)

• SOHCAHTOA is a mnemonic for the trigonometric ratios:
  • Sine (sin Θ) = Opposite / Hypotenuse
  • Cosine (cos Θ) = Adjacent / Hypotenuse
  • Tangent (tan Θ) = Opposite / Adjacent
• The other three functions are reciprocals:
  • Cosecant (csc Θ) = Hypotenuse / Opposite
  • Secant (sec Θ) = Hypotenuse / Adjacent
  • Cotangent (cot Θ) = Adjacent / Opposite

Special Right Triangles and Pythagorean Triplets

• Common triplets: 3-4-5, 5-12-13, 8-15-17, 7-24-25.
• Multiples of these triplets also form right triangles.
• Recognizing these helps quickly find missing sides.

Solving Right Triangles: Examples

• Use the Pythagorean Theorem to find the missing side.
• Find all six trigonometric functions using side values:
  • Sin Θ = Opposite / Hypotenuse
  • Cos Θ = Adjacent / Hypotenuse
  • Tan Θ = Opposite / Adjacent
  • Csc Θ = 1 / Sin Θ
  • Sec Θ = 1 / Cos Θ
  • Cot Θ = 1 / Tan Θ

Applying Trig Functions to Find Sides/Angles

• Use the appropriate trig function based on known sides/angles:
  • If given angle & adjacent/hypotenuse: use cosine.
  • If given angle & opposite/hypotenuse: use sine.
  • If given angle & opposite/adjacent: use tangent.
• To find a side, rearrange and solve for X.
• To find an angle, use the inverse function (arcsin, arccos, arctan).

Key Terms & Definitions

• Opposite — the side across from the given angle.
• Adjacent — the side next to the given angle (not the hypotenuse).
• Hypotenuse — the longest side, opposite the right angle.
• SOHCAHTOA — mnemonic for sine, cosine, tangent ratios.
• Reciprocal Trig Functions — csc, sec, cot; reciprocals of sin, cos, tan.
• Pythagorean Triplets — sets of three integers that satisfy a² + b² = c².

Action Items / Next Steps

• Review SOHCAHTOA and reciprocal identities.
• Practice finding missing sides and angles using trigonometric functions.
• Complete homework using provided right triangle problems.
• Preview additional topics: special triangles, laws of sines/cosines, trig identities, and equations.