Question

Find two negative and three positive angles, expressed in radians, for which the point on the unit circle that corresponds to each angle is $\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$.

Short answer

The angles are$\sin t=\frac{\sqrt{3}}{2},csct=\frac{2}{\sqrt{3}},\cos t=\frac{1}{2},\tan t=\sqrt{3}$

Step-by-step solution

Step 1. Given information .

Consider the given point$\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$.

Step 2. Finding negative and positive angles.

We follow the definition of six trigonometric function using $P\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)=\left(x,y\right)$where $x=\frac{1}{2},y=\frac{\sqrt{3}}{2}$.

Since the points are positive so there are positive angles.

$$\begin{gathered} \sin t=y=\frac{\sqrt{3}}{2},csct=\frac{1}{y}=\frac{2}{\sqrt{3}},\cos t=x=\frac{1}{2} \\ \tan t=\frac{y}{x}=\sqrt{3}, \end{gathered}$$