solving for values

Trigonometric Values Overview

  • Focus on right triangles in relation to trigonometric functions.

Right Triangle Sides

  • Three sides:

    • Adjacent: next to the angle

    • Opposite: across from the angle

    • Hypotenuse: opposite the right angle

Trigonometric Ratios

  • SOH CAH TOA mnemonic for remembering ratios:

    • Sine (sin) = Opposite (o) / Hypotenuse (h)

    • Cosine (cos) = Adjacent (a) / Hypotenuse (h)

    • Tangent (tan) = Opposite (o) / Adjacent (a)

Reciprocal Functions

  • Cosecant (csc) = Hypotenuse (h) / Opposite (o)

  • Secant (sec) = Hypotenuse (h) / Adjacent (a)

  • Cotangent (cot) = Adjacent (a) / Opposite (o)

Example Problems

  1. Finding Side Lengths:

    • Given hypotenuse of 16 units and angle of 50 degrees:

      • sin(50°) = Opposite / 16

      • cos(50°) = Adjacent / 16

    • Calculate:

      • Opposite = 16 * sin(50°) = 12.257 units

      • Adjacent = 16 * cos(50°) = 10.284 units

  2. Finding Cotangent:

    • Given sine(θ) = 2/5:

      • Use Pythagorean theorem: Opposite² + Adjacent² = Hypotenuse²

      • Calculate Adjacent side = √(25 - 4) = √21.

      • cot(θ) = Adjacent / Opposite = √21 / 2.

  3. Finding Trigonometric Functions for an Angle:

    • Given θ = -11π/3:

      • Find coterminal angle by adding multiples of 2π:

      • Computation leads to angle = π/3.

      • Sine(θ) = √3 / 2, Cosine(θ) = 1/2, Tangent(θ) = √3.

    • Compute reciprocal functions:

      • csc(θ) = 2√3 / 3, sec(θ) = 2, cot(θ) = √3 / 3.