Simplify square root of (1-cos(30))/2

Problem

√(,(1−cos(30))/2)

Solution

1. Identify the half-angle identity for sine, which states sin(θ/2)=√(,(1−cos(θ))/2) for angles in the first quadrant.

2. Substitute the value θ=30 into the identity to recognize that the expression represents sin(15)

3. Evaluate the cosine of 30 using the unit circle value cos(30)=√(,3)/2

4. Substitute the value back into the expression:

√(,(1−√(,3)/2)/2)

5. Simplify the numerator by finding a common denominator:

√(,(2−√(,3))/2/2)

6. Divide by 2 to simplify the fraction inside the square root:

√(,(2−√(,3))/4)

7. Apply the square root to the numerator and denominator separately:

√(,2−√(,3))/2

Final Answer

√(,(1−cos(30))/2)=√(,2−√(,3))/2

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