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

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

Options:

  1. √3/2
  2. 1/2
  3. 0
  4. √2/2

Correct Answer: √3/2

Solution: 

Using the identity sin^2(θ) + cos^2(θ) = 1, we have sin^2(θ) = 1 - (1/2)^2 = 1 - 1/4 = 3/4. Thus, sin(θ) = ±√3/2.

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

Practice Questions

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

Questions & Step-by-Step Solutions

If cos(θ) = 1/2, what is the value of sin(θ)?
  • Step 1: Start with the given value of cos(θ), which is 1/2.
  • Step 2: Use the Pythagorean identity, which states that sin^2(θ) + cos^2(θ) = 1.
  • Step 3: Substitute the value of cos(θ) into the identity: sin^2(θ) + (1/2)^2 = 1.
  • Step 4: Calculate (1/2)^2, which is 1/4.
  • Step 5: Rewrite the equation: sin^2(θ) + 1/4 = 1.
  • Step 6: To isolate sin^2(θ), subtract 1/4 from both sides: sin^2(θ) = 1 - 1/4.
  • Step 7: Calculate 1 - 1/4, which equals 3/4.
  • Step 8: Now we have sin^2(θ) = 3/4.
  • Step 9: To find sin(θ), take the square root of both sides: sin(θ) = ±√(3/4).
  • Step 10: Simplify √(3/4) to get sin(θ) = ±√3/2.
  • Trigonometric Identities – Understanding and applying the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 to find the sine value from the cosine value.
  • Quadrants of Trigonometric Functions – Recognizing that the sine function can be positive or negative depending on the angle's quadrant.
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