Find the Value Using the Unit Circle cot((5pi)/3)

Problem

cot((5*π)/3)

Solution

1. Identify the angle (5*π)/3 on the unit circle. This angle is in the fourth quadrant because (3*π)/2<(5*π)/3<2*π

2. Determine the coordinates (x,y) for the angle (5*π)/3 On the unit circle, these coordinates are (cos((5*π)/3),sin((5*π)/3))

3. Evaluate the cosine and sine values. For (5*π)/3 the reference angle is π/3 In the fourth quadrant, cosine is positive and sine is negative.

cos((5*π)/3)=1/2

sin((5*π)/3)=−√(,3)/2

4. Apply the formula for the cotangent function, which is the ratio of the x-coordinate to the y-coordinate.

cot(θ)=cos(θ)/sin(θ)

5. Substitute the values into the ratio.

cot((5*π)/3)=1/2/(−√(,3)/2)

6. Simplify the fraction by multiplying by the reciprocal.

cot((5*π)/3)=1/2⋅(−2/√(,3))

cot((5*π)/3)=−1/√(,3)

7. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

cot((5*π)/3)=−√(,3)/3

Final Answer

cot((5*π)/3)=−√(,3)/3

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