What is the exact value of sin(pi/3)?
Understand the Problem
The question is asking for the exact value of the sine function evaluated at pi/3 radians. This is a common question in trigonometry, which can be solved using the unit circle or standard trigonometric values.
Answer
The exact value is $\frac{\sqrt{3}}{2}$.
Answer for screen readers
The exact value of the sine function evaluated at $\frac{\pi}{3}$ radians is $\frac{\sqrt{3}}{2}$.
Steps to Solve
-
Identify known values of sine
Recall the exact values of sine for commonly used angles. The angle $\frac{\pi}{3}$ radians corresponds to $60^\circ$. -
Use the unit circle
On the unit circle, the coordinates at an angle of $\frac{\pi}{3}$ radians are $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. The sine function gives the y-coordinate of this point. -
Write the sine value
Thus, the value of the sine function at this angle is given by the y-coordinate:
$$ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} $$
The exact value of the sine function evaluated at $\frac{\pi}{3}$ radians is $\frac{\sqrt{3}}{2}$.
More Information
The sine of $\frac{\pi}{3}$ radians, or $60^\circ$, is a significant value in trigonometry and is commonly used in various applications such as physics and engineering.
Tips
- Confusing the sine values for different angles, particularly $\frac{\pi}{4}$ and $\frac{\pi}{6}$ with $\frac{\pi}{3}$.
To avoid this, always remember the exact values associated with common angles.
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