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Questions and Answers

What is the definition of sin(2x)?

  • sin²x + cos²x
  • 2sinxcosx (correct)
  • sinx + cosx
  • 2cos²x

What is one definition of cos(2x)?

  • cos²x - sin²x
  • 1 - 2sin²x
  • 2cos²x - 1
  • All of the above (correct)

What is the definition of tan(2x)?

2tanx / (1 - tan²x)

What is the definition of sin(x/2)?

<p>+-√(1 - cosx) / 2</p> Signup and view all the answers

What is the definition of cos(x/2)?

<p>+-√(1 + cosx) / 2</p> Signup and view all the answers

What is the definition of tan(x/2)?

<p>+-√(1 - cosx) / (1 + cosx)</p> Signup and view all the answers

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Study Notes

Half Angle and Double Angle Identities

  • sin(2x) is expressed as 2sinxcosx, representing the sine of double angles using the sine and cosine of the original angle.

  • cos(2x) can be represented by various forms:

    • cos²x - sin²x, demonstrating the relationship between squares of cosine and sine.
    • 2cos²x - 1, allowing transformation using cosine.
    • 1 - 2sin²x, providing an alternative expression based on sine.
  • tan(2x) is calculated using the formula 2tanx / (1 - tan²x), offering a tangent double angle expression derived from the tangent of the original angle.

  • For the half-angle identities:

    • sin(x/2) can be expressed as ±√(1-cosx)/2, relating the sine of halved angles to the cosine.

    • cos(x/2) is defined by ±√(1+cosx)/2, connecting the cosine of halved angles back to the original cosine value.

    • tan(x/2) is determined by ±√(1-cosx)/(1+cosx), showing the tangent of half angles as a function of the cosine.

These identities are crucial in trigonometry for simplifying expressions and solving equations.

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