Chapter 4.3, Problem 71E

Solution

To identify: The Drop perpendiculars from points P and Q to the x-axis to form two right triangles. Explain how the right triangles are related. refer to the unit circle in this figure. Point P is on the terminal side of an angle t, and point Q is on the terminal side of an angle $t+\pi /2$ .

  

The triangle $\Delta PORand\Delta QOS$ are congruent y side angle side.

Given information

The given refer to the unit circle in this figure. Point P is on the terminal side of an angle t, and point Q is on the terminal side of an angle $t+\pi /2$ .

  

Explanation:

Consider the given information

Circle is a unit circle so OP and OQ are radii of 1 unit

  

  

It is needed to determine relation between triangles $\Delta PORand\Delta QOS$

See in the triangle POR

  $\begin{array}{c}\angle POR=t\\ \angle OPR=\pi -\left(\angle POR+\angle OPR\right)\\ =\pi -\left(t-\frac{\pi }{2}\right)\\ =\frac{\pi }{2}-t\end{array}$

See in the triangle QOS

  $\begin{array}{c}\angle QOS=\pi -\left(\frac{\pi }{2}+t\right)\\ =\frac{\pi }{2}-t\end{array}$

Now

  $\begin{array}{c}\angle OQS=\pi -\left(\angle POR+\angle OPR\right)\\ =\pi -\left(\frac{\pi }{2}-t+\frac{\pi }{2}\right)\\ =t\end{array}$

Now in the triangle POR and QOS

  $\begin{array}{c}\angle POR=\angle OQS\\ \angle OPR=\angle QOS\\ \angle PRO=\angle QSO\\ PO=QO\end{array}$

Hence triangle $\Delta PORand\Delta QOS$ are congruent y side angle side.

So, the statement is true