What is the cotangent of pi/3?
Understand the Problem
The question is asking for the value of the cotangent function at the angle pi/3. This involves understanding the trigonometric function cotangent, which is the reciprocal of the tangent function, and using the known values of sine and cosine for this angle.
Answer
The value of $\cot\left(\frac{\pi}{3}\right)$ is $\frac{1}{\sqrt{3}}$.
Answer for screen readers
The value of $\cot\left(\frac{\pi}{3}\right)$ is $\frac{1}{\sqrt{3}}$.
Steps to Solve
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Recall the relationship of cotangent to tangent The cotangent of an angle is the reciprocal of the tangent of that angle. This can be expressed mathematically as: $$ \cot(x) = \frac{1}{\tan(x)} $$
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Find the tangent of the angle $\frac{\pi}{3}$ To find $ \tan\left(\frac{\pi}{3}\right)$, we can use known values from the unit circle. The coordinates for $\frac{\pi}{3}$ radians are: $$ \tan\left(\frac{\pi}{3}\right) = \frac{\sin\left(\frac{\pi}{3}\right)}{\cos\left(\frac{\pi}{3}\right)} $$ We know that: $$ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} $$ $$ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} $$
Substituting these values gives: $$ \tan\left(\frac{\pi}{3}\right) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} $$
- Calculate the cotangent of $\frac{\pi}{3}$ Now, we can find the cotangent: $$ \cot\left(\frac{\pi}{3}\right) = \frac{1}{\tan\left(\frac{\pi}{3}\right)} = \frac{1}{\sqrt{3}} $$
The value of $\cot\left(\frac{\pi}{3}\right)$ is $\frac{1}{\sqrt{3}}$.
More Information
The cotangent is a fundamental trigonometric function often used in right triangle calculations. Since $\frac{1}{\sqrt{3}}$ can also be rationalized to $\frac{\sqrt{3}}{3}$, both versions are commonly accepted.
Tips
- A common mistake is to confuse cotangent with tangent. Remember that cotangent is the reciprocal of tangent.
- Another mistake is to miscalculate the sine and cosine values for the given angle. Make sure to recall the correct values from the unit circle.
