Trigonometry Fundamentals
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Questions and Answers

What is the Pythagorean identity used for in solving right triangles?

  • To find the perimeter of the triangle
  • To find the area of the triangle
  • To find the length of the hypotenuse (correct)
  • To find the measure of an angle

    • What is the domain of the inverse sine function?

  • 0 ≤ θ ≤ π
  • -π ≤ θ ≤ π
  • -π/2 ≤ θ ≤ π/2 (correct)
  • -π/4 ≤ θ ≤ π/4

    • What is the period of the sine function?

  • π
  • π/2
  • 2π (correct)
  • 4π

    • What is the formula for the law of cosines?

    <p>c^2 = a^2 + b^2 - 2ab * cos(C)</p> Signup and view all the answers

    What is the application of trigonometry in modeling sound waves and light waves?

    <p>Wave applications</p> Signup and view all the answers

    What is the formula for the double angle formula for sine?

    <p>sin(2θ) = 2sin(θ)cos(θ)</p> Signup and view all the answers

    Study Notes

    Solving Triangles

    • Right Triangle Trigonometry:
      • Use sine, cosine, and tangent to relate angle measures to side lengths
      • Use Pythagorean identity: a^2 + b^2 = c^2 (c = hypotenuse)
    • Non-Right Triangle Trigonometry:
      • Use law of sines: a/sin(A) = b/sin(B) = c/sin(C)
      • Use law of cosines: c^2 = a^2 + b^2 - 2ab * cos(C)

    Inverse Trig Functions

    • Inverse Sine (arcsin):
      • Defined as: arcsin(x) = θ, where sin(θ) = x, -π/2 ≤ θ ≤ π/2
    • Inverse Cosine (arccos):
      • Defined as: arccos(x) = θ, where cos(θ) = x, 0 ≤ θ ≤ π
    • Inverse Tangent (arctan):
      • Defined as: arctan(x) = θ, where tan(θ) = x, -π/2 < θ < π/2
    • Identities:
      • arcsin(x) = arccos(√(1 - x^2))
      • arctan(x) = arccos(1/√(1 + x^2))

    Trigonometric Identities

    • Pythagorean Identity:
      • sin^2(θ) + cos^2(θ) = 1
    • Sum and Difference Formulas:
      • sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
      • cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
    • Double Angle Formulas:
      • sin(2θ) = 2sin(θ)cos(θ)
      • cos(2θ) = cos^2(θ) - sin^2(θ)

    Graphing Trigonometric Functions

    • Graph of Sine:
      • Period: 2π
      • Amplitude: 1
    • Graph of Cosine:
      • Period: 2π
      • Amplitude: 1
    • Graph of Tangent:
      • Period: π
      • Vertical asymptotes at π/2 + kπ (k ∈ ℤ)

    Applications of Trigonometry

    • Right Triangle Applications:
      • Height and distance problems
      • Angle of elevation and depression
    • Wave Applications:
      • Modeling sound waves and light waves
      • Analyzing periodic phenomena
    • Analytic Trigonometry:
      • Finding coordinates of points on a unit circle
      • Solving trigonometric equations using inverse functions

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    Description

    Test your understanding of trigonometry concepts, including right and non-right triangle trigonometry, inverse trig functions, identities, graphing, and applications. Evaluate your knowledge of sine, cosine, tangent, and more.

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