Trigonometry Formulas
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Questions and Answers

What is the value of sin(2A) if sin(A) = 1/2 and cos(A) = √3/2?

  • 1/4 (correct)
  • 1/2
  • √3/2
  • 1/√2
  • In a right triangle, if the length of the hypotenuse is 5 and the length of one leg is 3, what is the length of the other leg?

  • 4 (correct)
  • 2√2
  • 5
  • √13
  • If cos(A) = 2/3, what is the value of cos(A/2)?

  • ±√((4/9)/2)
  • ±√((1/9)/2)
  • ±√((5/9)/2) (correct)
  • ±√((7/9)/2)
  • What is the value of cos(2A) if sin(A) = 3/5 and cos(A) = 4/5?

    <p>13/25</p> Signup and view all the answers

    In an oblique triangle, if a = 5, b = 7, and C = 60°, what is the length of c?

    <p>9</p> Signup and view all the answers

    What is the value of sin(A + B) if sin(A) = 1/2, cos(A) = √3/2, sin(B) = 1/3, and cos(B) = 2/3?

    <p>7/12</p> Signup and view all the answers

    What is the value of tan(A) if sin(A) = 2/3 and cos(A) = √5/3?

    <p>2/√5</p> Signup and view all the answers

    Study Notes

    Trigonometric Identities

    • Pythagorean Identity: sin^2(A) + cos^2(A) = 1
    • Sum and Difference Identities:
      • sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
      • sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
      • cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
      • cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
    • Double Angle Identities:
      • sin(2A) = 2sin(A)cos(A)
      • cos(2A) = cos^2(A) - sin^2(A)
    • Half Angle Identities:
      • sin(A/2) = ±√((1 - cos(A))/2)
      • cos(A/2) = ±√((1 + cos(A))/2)

    Solving Triangles

    • Right Triangles:
      • SOH-CAH-TOA:
        • sine = opposite side / hypotenuse
        • cosine = adjacent side / hypotenuse
        • tangent = opposite side / adjacent side
      • Pythagorean Theorem: a^2 + b^2 = c^2 (where c is the hypotenuse)
    • Oblique Triangles:
      • Law of Sines: a / sin(A) = b / sin(B) = c / sin(C)
      • Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(C)
      • Heron's Formula: Area = √(s(s-a)(s-b)(s-c)) (where s is the semi-perimeter)
    • Solving Triangles Using Identities:
      • Use trigonometric identities to rewrite expressions in terms of a single angle or side
      • Simplify and solve for the desired angle or side

    Trigonometric Identities

    • Pythagorean Identity: relates the squares of sine and cosine of an angle to 1
    • Sum and Difference Identities: provide formulas for sine and cosine of sums and differences of angles
    • Double Angle Identities: relate the sine and cosine of double angles to products of sine and cosine of the original angle
    • Half Angle Identities: relate the sine and cosine of half angles to square roots of products of sine and cosine of the original angle

    Solving Triangles

    Right Triangles

    • SOH-CAH-TOA: a mnemonic for remembering the definitions of sine, cosine, and tangent in right triangles
    • Pythagorean Theorem: a formula for finding the length of the hypotenuse of a right triangle

    Oblique Triangles

    • Law of Sines: a formula for finding the lengths of sides or angles of an oblique triangle
    • Law of Cosines: a formula for finding the length of a side of an oblique triangle
    • Heron's Formula: a formula for finding the area of an oblique triangle

    Solving Triangles Using Identities

    • Rewriting Expressions: use trigonometric identities to rewrite expressions in terms of a single angle or side
    • Simplifying and Solving: simplify the rewritten expressions and solve for the desired angle or side

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