Calculating the Tangent Value of an Angle in the Unit Circle

Let’s consider an angle $ \theta $ in the unit circle. The coordinates of a point on the unit circle are given by $(\cos \theta, \sin \theta)$. The tangent of the angle $ \theta $ is defined as:

$\tan \theta = \frac{\sin \theta}{\cos \theta}$

Suppose $\theta = \frac{5\pi}{4}$, we need to find the value of $\tan \theta$. From the unit circle, we have:

$\sin \frac{5\pi}{4} = -\frac{\sqrt{2}}{2}$

$\cos \frac{5\pi}{4} = -\frac{\sqrt{2}}{2}$

Thus,

$\tan \frac{5\pi}{4} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1$

Consider an angle $ heta $ such that $ heta = frac{7pi}{6}$ on the unit circle. The coordinates of the point are $(cos heta, sin heta)$. The tangent value is:

$ an heta = frac{sin heta}{cos heta}$

For $ heta = frac{7pi}{6}$, we have:

$sin frac{7pi}{6} = -frac{1}{2}$

$cos frac{7pi}{6} = -frac{sqrt{3}}{2}$

Therefore,

$ an frac{7pi}{6} = frac{-frac{1}{2}}{-frac{sqrt{3}}{2}} = frac{1}{sqrt{3}} = frac{sqrt{3}}{3}$

Determine the value of $ an heta $ for $ heta = frac{3pi}{4}$.

The coordinates are:

$sin frac{3pi}{4} = frac{sqrt{2}}{2}$

$cos frac{3pi}{4} = -frac{sqrt{2}}{2}$

Therefore,

$ an frac{3pi}{4} = frac{frac{sqrt{2}}{2}}{-frac{sqrt{2}}{2}} = -1$

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