Find the value of $ sin( heta) $, $ cos( heta) $, and $ an( heta) $ for $ heta = frac{5pi}{6} $.

First, locate $ \theta = \frac{5\pi}{6} $ on the unit circle. This angle is in the second quadrant.

In the second quadrant, the sine function is positive and the cosine function is negative. The reference angle for $ \theta = \frac{5\pi}{6} $ is $ \frac{\pi}{6} $.

The sine and cosine values for $ \frac{\pi}{6} $ are $ \sin(\frac{\pi}{6}) = \frac{1}{2} $ and $ \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} $.

Since $ \theta = \frac{5\pi}{6} $ is in the second quadrant, we have:

$ \sin(\frac{5\pi}{6}) = \sin(\frac{\pi}{6}) = \frac{1}{2} $

$ \cos(\frac{5\pi}{6}) = -\cos(\frac{\pi}{6}) = -\frac{\sqrt{3}}{2} $

To find $ \tan(\frac{5\pi}{6}) $, use the identity $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $:

$ \tan(\frac{5\pi}{6}) = \frac{\sin(\frac{5\pi}{6})}{\cos(\frac{5\pi}{6})} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} $

Locate $ heta = frac{5pi}{6} $ on the unit circle, which lies in the second quadrant.

In the second quadrant, sine is positive and cosine is negative. The reference angle is $ frac{pi}{6} $.

We know:

$ sin(frac{pi}{6}) = frac{1}{2} $

$ cos(frac{pi}{6}) = frac{sqrt{3}}{2} $

Thus:

$ sin(frac{5pi}{6}) = frac{1}{2} $

$ cos(frac{5pi}{6}) = -frac{sqrt{3}}{2} $

For $ an(frac{5pi}{6}) $:

$ an(frac{5pi}{6}) = frac{sin(frac{5pi}{6})}{cos(frac{5pi}{6})} = frac{frac{1}{2}}{-frac{sqrt{3}}{2}} = -frac{1}{sqrt{3}} = -frac{sqrt{3}}{3} $

Given $ heta = frac{5pi}{6} $:

Quadrant: 2

Reference angle: $ frac{pi}{6} $

$ sin(frac{5pi}{6}) = frac{1}{2} $

$ cos(frac{5pi}{6}) = -frac{sqrt{3}}{2} $

$ an(frac{5pi}{6}) = -frac{sqrt{3}}{3} $

Start Using PopAi Today