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If x = tan^(-1)(√3), what is the value of sin(2x)?

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Question: If x = tan^(-1)(√3), what is the value of sin(2x)?

Options:

  1. √3/2
  2. 1
  3. √2/2
  4. 0

Correct Answer: 1

Solution: 

Since tan^(-1)(√3) = π/3, then 2x = 2π/3 and sin(2x) = sin(2π/3) = √3/2.

If x = tan^(-1)(√3), what is the value of sin(2x)?

Practice Questions

Q1
If x = tan^(-1)(√3), what is the value of sin(2x)?
  1. √3/2
  2. 1
  3. √2/2
  4. 0

Questions & Step-by-Step Solutions

If x = tan^(-1)(√3), what is the value of sin(2x)?
  • Step 1: Understand that x = tan^(-1)(√3) means we are looking for an angle whose tangent is √3.
  • Step 2: Recall that tan(π/3) = √3. Therefore, x = π/3.
  • Step 3: Calculate 2x by multiplying x by 2: 2x = 2 * (π/3) = 2π/3.
  • Step 4: Now, we need to find sin(2x), which is sin(2π/3).
  • Step 5: Recognize that sin(2π/3) is the same as sin(π - π/3).
  • Step 6: Use the sine identity: sin(π - θ) = sin(θ). So, sin(2π/3) = sin(π/3).
  • Step 7: Recall that sin(π/3) = √3/2.
  • Step 8: Therefore, sin(2x) = √3/2.
  • Inverse Trigonometric Functions – Understanding how to evaluate inverse trigonometric functions, specifically tan^(-1) in this case.
  • Trigonometric Identities – Applying the double angle formula for sine, sin(2x) = 2sin(x)cos(x), to find the value of sin(2x).
  • Unit Circle – Using the unit circle to determine the sine value of specific angles, such as 2π/3.
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