Right Triangle Trigonometry

Jul 18, 2024

Right Triangle Trigonometry

Introductory Concepts

  • SOHCAHTOA: A mnemonic device for remembering trig functions.
    • Sine ( ( heta) : Opposition to hypotenuse.
    • Cosine ( ( heta) : Adjacent to hypotenuse.
    • Tangent ( ( heta) : Opposition to adjacent.

Triangle Terminology

  • ** ( heta)** : Angle of focus.
  • Opposite side : Side opposite to ( heta).
  • Adjacent side : Side next to ( heta).
  • Hypotenuse : Side across the right angle, longest side.

Pythagorean Theorem

(a^2 + b^2 = c^2)

  • Used to find the missing side of a right triangle.

Trig Functions

  • Basic Trig Ratios: SOHCAHTOA
    • Sine ( ( heta) = Opposite / Hypotenuse
    • Cosine ( ( heta) = Adjacent / Hypotenuse
    • Tangent ( ( heta) = Opposite / Adjacent
  • Reciprocal Trig Functions
    • Cosecant ( ( heta) = 1 / Sine( ( heta)** )
    • Secant ( ( heta) = 1 / Cosine( ( heta)** )
    • Cotangent ( ( heta) = 1 / Tangent( ( heta)** )

Practical Application

Example 1

  • Given: Right triangle sides 3, 4, and hypotenuse.
  • Find the hypotenuse
    • Using Pythagorean theorem: 3^2 + 4^2 = 9 + 16 = 25. Hypotenuse = √25 = 5.
  • Find Trig Functions:
    • Sine ( ( heta) = Opposite / Hypotenuse = 4/5
    • Cosine ( ( heta) = Adjacent / Hypotenuse = 3/5
    • Tangent ( ( heta) = Opposite / Adjacent = 4/3
    • Cosecant ( ( heta) = 1 / Sine( ( heta)** ) = 5/4
    • Secant ( ( heta) = 1 / Cosine( ( heta)** ) = 5/3
    • Cotangent ( ( heta) = 1 / Tangent( ( heta)** ) = 3/4

Special Right Triangles

  • Common Triples:
    • 3-4-5
    • 5-12-13
    • 8-15-17
    • 7-24-25
  • Multiples of Triples Work:
    • Example: 6-8-10 (double 3-4-5), 9-12-15 (triple 3-4-5)

Example 2

  • Given: Right triangle sides 8, 17, and hypotenuse 15.
  • Find the hypotenuse
    • Using Pythagorean theorem: 8^2 + b^2 = 17^2.
    • Find b: 64 + b^2 = 289 → b^2 = 225 → b = 15.
  • Find Trig Functions:
    • Sine( ( heta)** = Opposite / Hypotenuse = 15/17
    • Cosine( ( heta)** = Adjacent / Hypotenuse = 8/17
    • Tangent( ( heta)** = Opposite / Adjacent = 15/8
    • Cosecant( ( heta)** = 1 / Sine( ( heta)**) = 17/15
    • Secant( ( heta)** = 1 / Cosine( ( heta)**) = 17/8
    • Cotangent( ( heta)** = 1 / Tangent( ( heta)**) = 8/15

Additional Problems and Concepts

  • Example with 32 degrees angle and given sides (use sine, cosine).
  • Using the calculator for tangent/cosine values.

Finding Missing Angles

  • Use inverse trig functions (arc functions).
    • Example: ( an^{-1}(5/4)\ = 51.34°)
    • Example: ( ext{arc cos}(3/7)\ = 64.62°)
    • Example: ( ext{arc sin}(5/6)\ = 56.44°)

Course Information

  • Udemy Trigonometry Course
    • Sections included: Angles, unit circle, right triangles, trigonometric functions of any angle, graphing, inverse trig functions, verifying identities, sum/difference formulas, solving trig equations, law of sines, law of cosines, polar coordinates.