Lecture Notes: Special Right Triangles (Part 2)

Key Points:

• Calculator Usage in Trigonometry:
  • Ensure calculator is in degree mode when working with degrees.
  • Most calculators have sine, cosine, and tangent functions. Secant is calculated as the reciprocal of cosine: sec(θ) = 1 / cos(θ).
  • Verify results using known properties: cosine should be < 1; secant should be > 1 (since hypotenuse > adjacent side).
  • Example: To find sec(54°), calculate 1 / cos(54°).

Special Right Triangles:

1. 45°-45°-90° Triangle:

• Also known as the isosceles right triangle.
• Ratios:
  • Leg : Leg : Hypotenuse = 1 : 1 : √2
  • If one leg is 5, hypotenuse = 5√2.
• Properties:
  • Angles opposite equal sides are congruent.
  • Hypotenuse is √2 times the length of a leg.

2. 30°-60°-90° Triangle:

• Ratios:
  • Short Leg : Long Leg : Hypotenuse = 1 : √3 : 2
  • Short leg = 1/2 hypotenuse.
  • Long leg = √3 times short leg.
• Properties:
  • 30° angle opposite shortest side.
  • 60° angle opposite longer leg.
  • 90° angle opposite hypotenuse.

Examples:

Example 1: 45°-45°-90° Triangle

• Given: Side = 5
• Solution:
  • Other leg = 5
  • Hypotenuse = 5√2

Example 2: 30°-60°-90° Triangle

• Given: Hypotenuse = 8
• Solution:
  • Shortest side = 4 (half of hypotenuse)
  • Long side = 4√3

Example 3: Solving a 30°-60°-90° Triangle

• Given: Long leg = 7
• Solution:
  • Short leg = 7/√3 (rationalize: 7√3/3)
  • Hypotenuse = Double short leg: 7√3

Word Problem: Ship Navigation

• Ship travels 53 mph northeast, forming a 30° angle with the positive x-axis.
• Total travel in 2 hours = 106 miles.
• Using 30°-60°-90° triangle ratios:
  • North (short leg) = 53 miles
  • East (long leg) = 53√3 ≈ 91.8 miles

Conclusion:

• Practice and familiarity with ratios are key to mastering these triangles.
• Continue practicing with different examples and verify calculations using properties of special right triangles.