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

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

Options:

  1. π/3
  2. π/2
  3. 2π/3
  4. π/4

Correct Answer: π/2

Solution: 

Using the identity cos^(-1)(√(1 - sin^2(x))) = π/2 - x, we find that cos^(-1)(√(1 - (1/3)^2)) = π/2 - sin^(-1)(1/3).

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

Practice Questions

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

Questions & Step-by-Step Solutions

If x = sin^(-1)(1/3), then what is the value of cos^(-1)(√(1 - (1/3)^2))?
  • Step 1: Understand that x = sin^(-1)(1/3) means that sin(x) = 1/3.
  • Step 2: Recall the identity that relates sine and cosine: cos^(-1)(√(1 - sin^2(x))) = π/2 - x.
  • Step 3: Calculate sin^2(x): (1/3)^2 = 1/9.
  • Step 4: Find 1 - sin^2(x): 1 - 1/9 = 8/9.
  • Step 5: Take the square root: √(8/9) = √8/3 = 2√2/3.
  • Step 6: Now apply the identity: cos^(-1)(√(1 - (1/3)^2)) = π/2 - sin^(-1)(1/3).
  • Step 7: Therefore, the value of cos^(-1)(√(1 - (1/3)^2)) is π/2 - x.
  • Inverse Trigonometric Functions – Understanding the relationships between inverse sine and cosine functions, particularly using identities.
  • Pythagorean Identity – Applying the identity that relates sine and cosine through the equation sin^2(x) + cos^2(x) = 1.
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