Question

The unit circle has been divided into twelve equal arcs, corresponding to t-values of

0, 𝜋/6, 𝜋/3, 𝜋/2, 2𝜋/3, 5𝜋/6, 𝜋, 7𝜋/6, 4𝜋/3, 3𝜋/2, 5𝜋/3, 11𝜋/6, and 2𝜋

Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
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cos 5𝜋/6

Accepted Answer

Recall that on the unit circle, the coordinates of a point corresponding to an angle $$t$$ are given by $$(\cos t, \sin t). $$Identify the angle $$t = \frac{5\pi}{6} on $$the unit circle. This angle is located in the second quadrant, where cosine values are negative and sine values are positive. Use the known exact coordinates for $$\frac{5\pi}{6}. $$Since the circle is divided into twelve equal arcs of $$\frac{\pi}{6}$$ each, the coordinates at $$\frac{5\pi}{6}$$ correspond to the point $$\left(-\cos \frac{\pi}{6}, \sin \frac{\pi}{6}\right). $$Recall the exact values: $$\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$$ and $$\sin \frac{\pi}{6} = \frac{1}{2}. $$Substitute these into the coordinates to get $$\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right). $$Since cosine corresponds to the x-coordinate on the unit circle, the value of $$\cos \frac{5\pi}{6} is $$the x-coordinate of this point, which is $$-\frac{\sqrt{3}}{2}.$$

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Key Concepts

Unit Circle and Radian Measure

Coordinates on the Unit Circle and Trigonometric Functions

Domain and Undefined Values of Trigonometric Functions