What is the value of cos(2π over 3) ? Use your Unit Circle to help you answer this question.(1 point)

13 answers

To find the value of \(\cos\left(\frac{2\pi}{3}\right)\), we can use the unit circle.

The angle \(\frac{2\pi}{3}\) radians is in the second quadrant. We can find it by recognizing that \(\frac{2\pi}{3}\) is equivalent to \(120^\circ\), which is calculated by converting radians to degrees using the formula:

\[ \frac{2\pi}{3} \cdot \frac{180^\circ}{\pi} = 120^\circ \]

On the unit circle, the coordinates for the angle \(120^\circ\) or \(\frac{2\pi}{3}\) are:

\[ \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \]

The cosine value corresponds to the x-coordinate of the point on the unit circle. Therefore,

\[ \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \]

Thus, the value of \(\cos\left(\frac{2\pi}{3}\right)\) is \(-\frac{1}{2}\).

Did the unit circle help

Yes, the unit circle is essential in determining the values of trigonometric functions like cosine.

Here's how the unit circle helps in this case:

  1. Identify the Angle: We found that \(\frac{2\pi}{3}\) radians corresponds to \(120^\circ\), which is in the second quadrant of the unit circle.

  2. Locate the Point: In the second quadrant, the coordinates of the point on the unit circle corresponding to \(120^\circ\) (or \(\frac{2\pi}{3}\)) are \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\).

  3. Extract Cosine Value: The x-coordinate of this point, which represents the cosine of the angle, is \(-\frac{1}{2}\). Therefore, we conclude that \(\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\).

In summary, the unit circle visually and numerically aids in understanding and finding trigonometric function values for given angles.

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