Find the Values of $ an( heta)$ Given a Point on the Unit Circle

Answer 1

Given a point $P(a, b)$ on the unit circle, find the value of $\tan(\theta)$ where $\theta$ is the angle formed by the radius to the point P and the positive x-axis.

Solution: We know that $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. Since P(a, b) lies on the unit circle, $\cos(\theta) = a$ and $\sin(\theta) = b$. Therefore, $\tan(\theta) = \frac{b}{a}$.

So, the value of $\tan(\theta)$ is $\frac{b}{a}$.

Answer 2

Given a point $P(frac{1}{2}, -frac{sqrt{3}}{2})$ on the unit circle, find the value of $ an( heta)$ where $ heta$ is the angle formed by the radius to the point P and the positive x-axis.

Solution: We know that $ an( heta) = frac{sin( heta)}{cos( heta)}$. Since P(frac{1}{2}, -frac{sqrt{3}}{2}) lies on the unit circle, $cos( heta) = frac{1}{2}$ and $sin( heta) = -frac{sqrt{3}}{2}$. Therefore, $ an( heta) = frac{-frac{sqrt{3}}{2}}{frac{1}{2}} = -sqrt{3}$.

So, the value of $ an( heta)$ is $-sqrt{3}$.

Answer 3

Given a point $P(-frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$ on the unit circle, find the value of $ an( heta)$ where $ heta$ is the angle formed by the radius to the point P and the positive x-axis.

Solution: We know that $ an( heta) = frac{sin( heta)}{cos( heta)}$. Since P(-frac{sqrt{2}}{2}, frac{sqrt{2}}{2}) lies on the unit circle, $cos( heta) = -frac{sqrt{2}}{2}$ and $sin( heta) = frac{sqrt{2}}{2}$. Therefore, $ an( heta) = frac{frac{sqrt{2}}{2}}{-frac{sqrt{2}}{2}} = -1$.

So, the value of $ an( heta)$ is $-1$.

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