Chapter 13.2, Problem 18PPS

To determine

To find : $Are:(\theta )=\frac{7\sin \theta +5\cos \theta }{\sin \theta \cos \theta }and\text{l}(\theta )=7\sec \theta +5\mathrm{cosec}\theta \text{ equivalent ?}$

Explanation of Solution

Given information :

Student derived an expression for length of ladder when carried flat, could fit around a corner from a $5$ -foot wide hallway into a $7$ -foot-wide hallway, maximum length $l$ of a ladder given by $\text{l}(\theta )=\frac{7\sin \theta +5\cos \theta }{\sin \theta \cos \theta }$ , for same teacher concluded $\text{l}(\theta )=7\sec \theta +5\mathrm{cosec}\theta$ .

Calculation :

  $\begin{array}{l}\frac{7\sin \theta +5\cos \theta }{\sin \theta \cos \theta }=7\sec \theta +5\mathrm{cosec}\theta \\ \frac{7\sin \theta }{\sin \theta \cos \theta }+\frac{5\cos \theta }{\sin \theta \cdot \cos \theta }=7\sec \theta +5\mathrm{cosec}\theta \\ \frac{7}{\cos \theta }+\frac{5}{\sin \theta }=7\sec \theta +5\mathrm{cosec}\theta \\ 7\sec \theta +5\mathrm{cosec}\theta =7\sec \theta +5\mathrm{cosec}\theta \end{array}$

Yes, They are equivalent.