Find the Exact Values of Trigonometric Functions for Given Angles

Given the angle $\theta = \frac{5\pi}{6}$, find the exact values of $\sin \theta$, $\cos \theta$, and $\tan \theta$ using the unit circle.

First, determine the reference angle for $\theta = \frac{5\pi}{6}$. Since $\frac{5\pi}{6}$ lies in the second quadrant, its reference angle is:

$\pi – \frac{5\pi}{6} = \frac{\pi}{6}$

The sine, cosine, and tangent values for $\frac{\pi}{6}$ are:

$\sin \frac{\pi}{6} = \frac{1}{2}, \quad \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}, \quad \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}$

In the second quadrant, sine is positive while cosine and tangent are negative. Therefore:

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

Given the angle $ heta = frac{5pi}{6}$, find the exact values of $sin heta$, $cos heta$, and $ an heta$ using the unit circle.

The angle $ heta = frac{5pi}{6}$ lies in the second quadrant, where sine is positive and cosine is negative. The reference angle $alpha$ for $ heta = frac{5pi}{6}$ is:

$alpha = pi – frac{5pi}{6} = frac{pi}{6}$

Using the known values from the unit circle:

$sin frac{pi}{6} = frac{1}{2}, quad cos frac{pi}{6} = frac{sqrt{3}}{2}, quad an frac{pi}{6} = frac{1}{sqrt{3}}$

We find the exact values for $ heta = frac{5pi}{6}$ to be:

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

Given $ heta = frac{5pi}{6}$, find $sin heta$, $cos heta$, and $ an heta$ using the unit circle.

In the second quadrant, reference angle:

$pi – frac{5pi}{6} = frac{pi}{6}$

Values:

$sin frac{pi}{6} = frac{1}{2}, quad cos frac{pi}{6} = frac{sqrt{3}}{2}, quad an frac{pi}{6} = frac{1}{sqrt{3}}$

Using unit circle properties:

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

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