Find the Exact Value cot(15)

Problem

cot(15)

Solution

1. Identify the angle as a difference of two special angles from the unit circle, such as 45 and 30

cot(15)=cot(45−30)

2. Apply the formula for the cotangent of a difference, which is cot(A−B)=(cot(A)*cot(B)+1)/(cot(B)−cot(A))

cot(45−30)=(cot(45)*cot(30)+1)/(cot(30)−cot(45))

3. Substitute the known values cot(45)=1 and cot(30)=√(,3) into the expression.

((1)*(√(,3))+1)/(√(,3)−1)

4. Simplify the expression to obtain the fraction.

(√(,3)+1)/(√(,3)−1)

5. Rationalize the denominator by multiplying the numerator and the denominator by the conjugate √(,3)+1

((√(,3)+1)*(√(,3)+1))/((√(,3)−1)*(√(,3)+1))

6. Expand the numerator and denominator.

(3+2√(,3)+1)/(3−1)

7. Simplify the resulting fraction.

(4+2√(,3))/2

8. Divide each term in the numerator by the denominator.

2+√(,3)

Final Answer

cot(15)=2+√(,3)

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