Trigonometric Identities and Triangles
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Questions and Answers

What is the formula for tan(A + B)?

  • (tan(A) + tan(B)) / (1 + tan(A)tan(B))
  • (tan(A) + tan(B)) / (1 - tan(A)tan(B)) (correct)
  • (tan(A) - tan(B)) / (1 + tan(A)tan(B))
  • (tan(A) - tan(B)) / (1 - tan(A)tan(B))

    • What is the reference angle for an angle in standard position?

  • The hypotenuse of the right triangle
  • The acute angle between the terminal side and the x-axis (correct)
  • The angle between the initial side and the terminal side
  • The right triangle adjacent to the terminal side

    • What is the formula for cos(2A)?

  • 2cos²(A) - 1
  • cos²(A) + sin²(A)
  • cos²(A) - sin²(A) (correct)
  • 1 - 2sin²(A)

    • What is the sine of an angle A?

    <p>Opposite side / Hypotenuse</p> Signup and view all the answers

    What is the Pythagorean identity?

    <p>sin²(A) + cos²(A) = 1</p> Signup and view all the answers

    What is the formula for sin(A + B)?

    <p>sin(A)cos(B) + cos(A)sin(B)</p> Signup and view all the answers

    Study Notes

    Trigonometric Identities

    • Pythagorean Identity: sin²(A) + cos²(A) = 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)
      • tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
    • Double Angle Formulas:
      • sin(2A) = 2sin(A)cos(A)
      • cos(2A) = cos²(A) - sin²(A)
      • tan(2A) = (2tan(A)) / (1 - tan²(A))

    Triangles And Angles

    • Right Triangle: one angle is 90° (π/2 radians)
    • Angles in Standard Position:
      • Initial side: x-axis (positive)
      • Terminal side: where the angle ends (can be any quadrant)
    • Reference Angle:
      • The acute angle between the terminal side and the x-axis
      • Used to find trig values for non-acute angles

    Sine Cosine And Tangent

    • Sine (sin):
      • Opposite side / Hypotenuse
      • sin(A) = opposite side / hypotenuse
    • Cosine (cos):
      • Adjacent side / Hypotenuse
      • cos(A) = adjacent side / hypotenuse
    • Tangent (tan):
      • Opposite side / Adjacent side
      • tan(A) = opposite side / adjacent side

    Trigonometric Identities

    • The Pythagorean Identity is a fundamental concept in trigonometry, stating that sin²(A) + cos²(A) = 1.
    • The Sum and Difference Formulas describe the trigonometric relationships between angles A and B:
      • sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
      • cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
      • tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
    • The Double Angle Formulas provide trigonometric values for double angles:
      • sin(2A) = 2sin(A)cos(A)
      • cos(2A) = cos²(A) - sin²(A)
      • tan(2A) = (2tan(A)) / (1 - tan²(A))

    Triangles and Angles

    • A right triangle has one angle measuring 90° (π/2 radians).
    • Angles in standard position have their initial side on the x-axis (positive) and their terminal side in any quadrant.
    • The reference angle is the acute angle between the terminal side and the x-axis, used to find trigonometric values for non-acute angles.

    Sine, Cosine, and Tangent

    • Sine (sin) is defined as the opposite side divided by the hypotenuse: sin(A) = opposite side / hypotenuse.
    • Cosine (cos) is defined as the adjacent side divided by the hypotenuse: cos(A) = adjacent side / hypotenuse.
    • Tangent (tan) is defined as the opposite side divided by the adjacent side: tan(A) = opposite side / adjacent side.

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    Description

    Test your understanding of trigonometric identities, including the Pythagorean identity, sum and difference formulas, and double angle formulas, as well as topics related to triangles and angles.

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