Question
Find the exact value of the expression. \( \csc \left(\frac{\pi}{3}\right)= \)
Ask by Mcfarlane Ross. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The exact value of \( \csc \left(\frac{\pi}{3}\right) \) is \( \frac{2\sqrt{3}}{3} \).
Solution
Calculate or simplify the expression \( \csc(\frac{\pi}{3}) \).
Calculate the value by following steps:
- step0: Calculate:
\(\csc\left(\frac{\pi }{3}\right)\)
- step1: Calculate the trigonometric value:
\(\frac{2\sqrt{3}}{3}\)
The exact value of \( \csc \left(\frac{\pi}{3}\right) \) is \( \frac{2\sqrt{3}}{3} \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
The cosecant function is the reciprocal of the sine function. We know that \( \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \). Thus, to find \( \csc \left( \frac{\pi}{3} \right) \), we take the reciprocal of \( \sin \left( \frac{\pi}{3} \right) \): \[ \csc \left( \frac{\pi}{3} \right) = \frac{1}{\sin \left( \frac{\pi}{3} \right)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \] To make it neater, we can rationalize the denominator: \[ \csc \left( \frac{\pi}{3} \right) = \frac{2 \cdot \sqrt{3}}{3} \] So, the exact value of \( \csc \left( \frac{\pi}{3} \right) \) is \( \frac{2 \sqrt{3}}{3} \).
