Find the Exact Value cot(420)

Problem

cot(420)

Solution

1. Find the coterminal angle by subtracting 360 from the given angle to find an equivalent angle within the first rotation.

420−360=60

2. Apply the periodic property of the cotangent function, which states cot(θ+360*k)=cot(θ)

cot(420)=cot(60)

3. Use the trigonometric definition of cotangent in terms of sine and cosine.

cot(60)=cos(60)/sin(60)

4. Substitute the known values for the sine and cosine of 60 from the unit circle.

cos(60)=1/2

sin(60)=√(,3)/2

5. Simplify the fraction by dividing the two values.

(1/2)/(√(,3)/2)=1/√(,3)

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

1/√(,3)⋅√(,3)/√(,3)=√(,3)/3

Final Answer

cot(420)=√(,3)/3

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