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

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

Options:

  1. 1/2
  2. √2/2
  3. √3/2
  4. 1

Correct Answer: √2/2

Solution: 

If x = sin^(-1)(1/√2), then sin(x) = 1/√2. Therefore, cos(x) = √(1 - sin^2(x)) = √(1 - (1/√2)^2) = √(1/2) = √2/2.

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

Practice Questions

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

Questions & Step-by-Step Solutions

If x = sin^(-1)(1/√2), then what is the value of cos(x)?
Correct Answer: √2/2
  • Step 1: Understand that x = sin^(-1)(1/√2) means that sin(x) = 1/√2.
  • Step 2: Recall the Pythagorean identity which states that sin^2(x) + cos^2(x) = 1.
  • Step 3: Substitute sin(x) into the identity: sin^2(x) = (1/√2)^2.
  • Step 4: Calculate (1/√2)^2, which equals 1/2.
  • Step 5: Now, use the identity: 1/2 + cos^2(x) = 1.
  • Step 6: Rearrange the equation to find cos^2(x): cos^2(x) = 1 - 1/2.
  • Step 7: Calculate 1 - 1/2, which equals 1/2.
  • Step 8: Now, take the square root to find cos(x): cos(x) = √(1/2).
  • Step 9: Simplify √(1/2) to get √2/2.
  • Inverse Trigonometric Functions – Understanding the relationship between sine and cosine through inverse functions.
  • Pythagorean Identity – Using the identity sin^2(x) + cos^2(x) = 1 to find cosine from sine.
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