LM

Math127 Right Triangle Trig

Right Triangle Trigonometry

Definition

  • Right Triangle Trigonometry involves the relationships between the angles and sides of a right triangle.

  • Key Sides:

    • Hypotenuse: The longest side, opposite the right angle.

    • Opposite Side: The side opposite the given angle.

    • Adjacent Side: The side next to the given angle.

Trigonometric Functions

  • Primary Trigonometric Functions:

    • Sine (sin θ): Opposite / Hypotenuse

    • Cosine (cos θ): Adjacent / Hypotenuse

    • Tangent (tan θ): Opposite / Adjacent

Mnemonics for Remembering Functions

  • Letter Method:

    • Use the first letters of Sine, Cosine, Tangent (S, C, T) to recall which sides correspond to which functions.

  • SOHCAHTOA Acronym:

    • SOH: Sine = Opposite / Hypotenuse

    • CAH: Cosine = Adjacent / Hypotenuse

    • TOA: Tangent = Opposite / Adjacent

Finding Cosecant, Secant, and Cotangent

  • Reciprocal Functions:

    • Cosecant (csc θ): Hypotenuse / Opposite (1/sin θ)

    • Secant (sec θ): Hypotenuse / Adjacent (1/cos θ)

    • Cotangent (cot θ): Adjacent / Opposite (1/tan θ)

Angle Values and Their Trigonometric Functions

  • Important angles to memorize:

    • 30° (π/6): Opposite = 1, Adjacent = √3, Hypotenuse = 2

    • 45° (π/4): Opposite = 1, Adjacent = 1, Hypotenuse = √2

    • 60° (π/3): Opposite = √3, Adjacent = 1, Hypotenuse = 2

Example Right Triangles

  • 30-degree Right Triangle (π/6):

    • Opposite = 1, Hypotenuse = 2, Adjacent = √3

    • Ratio of sides:

      • Sine = 1/2

      • Cosine = √3/2

      • Tangent = 1/√3

  • 45-degree Right Triangle (π/4):

    • Both opposite and adjacent sides are equal:

      • Sine = 1/√2

      • Cosine = 1/√2

      • Tangent = 1

  • 60-degree Right Triangle (π/3):

    • Opposite = √3, Hypotenuse = 2, Adjacent = 1

    • Sine = √3/2, Cosine = 1/2, Tangent = √3

Cofunction Identities

  • Relationships Between Sine, Cosine, and Other Functions:

    • sine θ = cosine (90° - θ)

    • cosine θ = sine (90° - θ)

    • tangent θ = cotangent (90° - θ)

    • cosecant θ = secant (90° - θ)

Example Calculations

  1. Finding Six Trigonometric Functions:

    • Redraw the triangle for clarity.

    • Example Triangle: Opposite = 6, Adjacent = 8, Hypotenuse = 10.

    • Calculate:

      • Sine = 6/10 = 3/5

      • Cosine = 8/10 = 4/5

      • Tangent = 6/8 = 3/4

      • Cosecant = 10/6 = 5/3

      • Secant = 10/8 = 5/4

      • Cotangent = 8/6 = 4/3

  2. Using the Pythagorean Theorem:

    • To find a missing side, use: Hypotenuse² = Opposite² + Adjacent²

    • Example: Given Opposite = 5, Adjacent = 12, find Hypotenuse = √(5² + 12²) = 13.

Final Notes on Angle Values

  • Important values for:

    • 90° (π/2): Sine = 1, Cosine = 0

    • 180° (π): Sine = 0, Cosine = -1

    • 270° (3π/2): Sine = -1, Cosine = 0

    • 360° (2π): Sine = 0, Cosine = 1

Review and Memorization Methods

  • Either memorize specific values for angles or visual shapes of triangles.

  • Utilizing charts and ratios for quick reference can aid in better understanding and recalling trigonometric relationships during problem-solving.