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If x = cos^(-1)(1/2), what is the value of sin(x)?

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Question: If x = cos^(-1)(1/2), what is the value of sin(x)?

Options:

  1. √3/2
  2. 1/2
  3. 0
  4. 1

Correct Answer: √3/2

Solution: 

Using the identity sin(x) = sqrt(1 - cos^2(x)), we have sin(x) = sqrt(1 - (1/2)^2) = √3/2.

If x = cos^(-1)(1/2), what is the value of sin(x)?

Practice Questions

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

Questions & Step-by-Step Solutions

If x = cos^(-1)(1/2), what is the value of sin(x)?
Correct Answer: √3/2
  • Step 1: Understand that x = cos^(-1)(1/2) means we are looking for an angle x whose cosine value is 1/2.
  • Step 2: Recall that cos(x) = 1/2 corresponds to the angle x = 60 degrees or x = Ο€/3 radians.
  • Step 3: Use the identity sin(x) = sqrt(1 - cos^2(x)) to find sin(x).
  • Step 4: Calculate cos^2(x): Since cos(x) = 1/2, then cos^2(x) = (1/2)^2 = 1/4.
  • Step 5: Substitute cos^2(x) into the identity: sin(x) = sqrt(1 - 1/4).
  • Step 6: Simplify the expression: 1 - 1/4 = 3/4, so sin(x) = sqrt(3/4).
  • Step 7: Further simplify: sin(x) = sqrt(3)/sqrt(4) = sqrt(3)/2.
  • Inverse Trigonometric Functions – Understanding how to find angles using inverse trigonometric functions, specifically cos^(-1) in this case.
  • Pythagorean Identity – Applying the identity sin^2(x) + cos^2(x) = 1 to find the sine value from the cosine value.
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