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

Want more problems? Check here!