Find the exact value of cot(π)/6 in simplest form with a rational denominator.
The Correct Answer and Explanation is:
To find the exact value of:cot(π6)\cot\left(\frac{\pi}{6}\right)cot(6π)
Step 1: Understand the cotangent function
The cotangent function is defined as:cot(θ)=1tan(θ)=cos(θ)sin(θ)\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}cot(θ)=tan(θ)1=sin(θ)cos(θ)
So,cot(π6)=cos(π6)sin(π6)\cot\left(\frac{\pi}{6}\right) = \frac{\cos\left(\frac{\pi}{6}\right)}{\sin\left(\frac{\pi}{6}\right)}cot(6π)=sin(6π)cos(6π)
Step 2: Use known values for special angles
From the unit circle:
- cos(π6)=32\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}cos(6π)=23
- sin(π6)=12\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}sin(6π)=21
Now substitute into the formula:cot(π6)=3212=3\cot\left(\frac{\pi}{6}\right) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}cot(6π)=2123=3
Final Answer:
3\boxed{\sqrt{3}}3
Explanation
The cotangent function is one of the six fundamental trigonometric functions and is the reciprocal of the tangent function. For any angle θ\thetaθ, the cotangent is defined as the ratio of the cosine of the angle to the sine of the angle. This makes the formula:cot(θ)=cos(θ)sin(θ)\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}cot(θ)=sin(θ)cos(θ)
In this problem, we are asked to evaluate cot(π6)\cot\left(\frac{\pi}{6}\right)cot(6π). This is a standard angle in trigonometry, equivalent to 30 degrees. It is useful to recall the exact values of sine and cosine at this angle from memory or a unit circle chart. The cosine of π6\frac{\pi}{6}6π is 32\frac{\sqrt{3}}{2}23, and the sine is 12\frac{1}{2}21. By applying the cotangent definition, we divide the cosine by the sine:3212\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}2123
This is a complex fraction, but it simplifies easily. When dividing two fractions, you multiply the numerator by the reciprocal of the denominator:32×21=3\frac{\sqrt{3}}{2} \times \frac{2}{1} = \sqrt{3}23×12=3
This result is already in its simplest form, and the denominator is rational. Therefore, the exact value of cot(π6)\cot\left(\frac{\pi}{6}\right)cot(6π) is 3\sqrt{3}3, a positive irrational number that is frequently encountered in trigonometric problems.
