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If x = sin^(-1)(1/3), then what is the value of cos^(-1)(√(1 - (1/3)^2))?
If x = sin^(-1)(1/3), then what is the value of cos^(-1)(√(1 - (1/3)^2))?
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Practice Questions
1 question
Q1
If x = sin^(-1)(1/3), then what is the value of cos^(-1)(√(1 - (1/3)^2))?
π/3
π/2
2π/3
π/4
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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).
Questions & Step-by-step Solutions
1 item
Q
Q: If x = sin^(-1)(1/3), then what is the value of cos^(-1)(√(1 - (1/3)^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).
Steps: 7
Show Steps
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.
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