Find the Exact Value cot(75)

Problem

cot(75)

Solution

1. Identify the angle as a sum of two special angles from the unit circle.

75=45+30

2. Apply the formula for the cotangent of a sum, 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 for cot(45)=1 and cot(30)=√(,3)

cot(75)=((1)*(√(,3))−1)/(√(,3)+1)

4. Simplify the expression by rationalizing the denominator. Multiply the numerator and denominator by the conjugate √(,3)−1

cot(75)=(√(,3)−1)/(√(,3)+1)⋅(√(,3)−1)/(√(,3)−1)

5. Expand the numerator and denominator.

cot(75)=(3−√(,3)−√(,3)+1)/(3−1)

6. Combine like terms and simplify the fraction.

cot(75)=(4−2√(,3))/2

cot(75)=2−√(,3)

Final Answer

cot(75)=2−√(,3)

Want more problems? Check here!