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.