cot of pi/3
Understand the Problem
The question is asking for the value of the cotangent of the angle pi/3. The cotangent function is the reciprocal of the tangent function, and this will involve evaluating the tangent of pi/3 and then taking the reciprocal of that value.
Answer
$\frac{\sqrt{3}}{3}$
Answer for screen readers
The value of the cotangent of the angle $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{3}$.
Steps to Solve
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Identify the Angle We are tasked with finding the cotangent of the angle $\frac{\pi}{3}$.
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Evaluate the Tangent Function The cotangent function is the reciprocal of the tangent function. We first need to find $\tan\left(\frac{\pi}{3}\right)$. The tangent of $\frac{\pi}{3}$ can be evaluated using the sine and cosine:
$$ \tan\left(\frac{\pi}{3}\right) = \frac{\sin\left(\frac{\pi}{3}\right)}{\cos\left(\frac{\pi}{3}\right)} $$
- Find Sine and Cosine Values The sine and cosine values for the angle $\frac{\pi}{3}$ are:
$$ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} $$
$$ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} $$
Using these values, we calculate:
$$ \tan\left(\frac{\pi}{3}\right) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} $$
- Calculate Cotangent Now we calculate cotangent by taking the reciprocal of the tangent:
$$ \cot\left(\frac{\pi}{3}\right) = \frac{1}{\tan\left(\frac{\pi}{3}\right)} = \frac{1}{\sqrt{3}} $$
- Rationalize the Denominator To express this in a more standard form, we multiply the numerator and denominator by $\sqrt{3}$:
$$ \cot\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{3} $$
The value of the cotangent of the angle $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{3}$.
More Information
The cotangent function, being the reciprocal of tangent, shows how relationships between angles and sides in a right triangle can yield different functions. The value of cotangent is particularly useful in trigonometry, especially in solving various geometric problems.
Tips
A common mistake is forgetting that cotangent is the reciprocal of tangent. Make sure to calculate tangent correctly first before taking the reciprocal.
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