Question

Suppose that the terminal point determined by $t$is the point $\left(\frac{3}{5},\frac{4}{5}\right)$on the unit circle. Find the terminal point determined by the following point.

$\frac{\pi }{2}-t$

Short answer

The terminal point for the given angle is $\left(\text{0.8},0.6\right)$.

Step-by-step solution

Given information.

The given point on the unit circle is $\left(\frac{3}{5},\frac{4}{5}\right)$.

Formula used.

The formula to find the angle is defined as,

$\tan \theta =\frac{y}{x}$

Calculation.

Consider the given point,$\left(\frac{3}{5},\frac{4}{5}\right)$

Compare the given point with $\left(x,y\right)$.

$x=\frac{3}{5},y=\frac{4}{5}$

Substitute the values in the formula.

role="math" localid="1648561214666" $$\begin{gathered} \tan \theta =\frac{\frac{3}{5}}{\frac{4}{5}} \\ \theta ={\tan }^{-1}\left(\frac{3}{4}\right) \\ =\text{36.87}° \end{gathered}$$

The required angle can be calculated as,

$$\begin{gathered} \theta =\frac{\pi }{2}-t \\ =90°-36.87° \\ =53.13° \end{gathered}$$

Now, substitute the value of angle in terminal point $\left(x\text{'},y\text{'}\right)=\left(\sin \theta ,\cos \theta \right)$.

role="math" localid="1648561326998" $$\begin{gathered} \left(x\text{'},y\text{'}\right)=\left(\sin 53.13°,\cos 53.13°\right) \\ =\left(\text{0.8},0.6\right) \end{gathered}$$