Chapter 13.1, Problem 45HP

To determine

To find: Why ${\cos }^2\text{θ}+{\sin }^2\text{θ}=\text{1}$ Ps classified in this way?

Answer to Problem 45HP

Because , it gives all relationship between angles and sides on a right angle.

Explanation of Solution

Given information:

We know that, Pythagoras theorem and ${\cos }^2\text{θ}+{\sin }^2\text{θ}=\text{1}$

Calculation:

In $\Delta ABC$ , using Pythagoras,

  $\sin \theta =\frac{c}{b},\cos \theta =\frac{a}{b}$

  $a^2+c^2=b^2$ (from Pythagoras)

  $\begin{array}{l}c=b\sin \theta ,a=b\cos \theta \\ $ {(b\cos \theta )}^2+{(b\sin \theta )}^2=b^2$b^2{\cos }^2\theta +b^2{\cos }^2\theta =b^2\end{array}$

Dividing by $b^2$

  $\begin{array}{l}\frac{b^2{\cos }^2\theta }{b^2}+\frac{b^2{\sin }^2\theta }{b^2}=b^2\\ {\cos }^2\theta +{\sin }^2\theta =1\end{array}$

It hold important relation between sides and angles of right triangle, therefore this way.