Chapter 8.6, Problem 26WE

To determine

To Prove: The value of k in respect of sin and cos.

Explanation of Solution

Given information:

A rectangular card is 10 cm wide.

  

Consider the figure;

  

Apply the cosine formula in the right angle triangle,

  $\begin{array}{l}\begin{array}{cc}\cos n^{\circ }=& \frac{10}{C{D}^{\prime }}\\ \cos n^{\circ }=& \frac{C{D}^{\prime }}{k}\\ \cos n^{\circ }=& \frac{10}{k\cos n^{\circ }}\end{array}\\ k{\cos }^2{n}^{\circ }=10\end{array}$

Combining the value of sine and cosine to get

  $\begin{array}{l}\begin{array}{r}\hfill 2\sin n^{\circ }\cos n^{\circ }\cos n^{\circ }\times k=10\\ \hfill k\sin (2n)\cos n^{\circ }=10\end{array}\\ k=\frac{10}{\sin {(2n)}^{\circ }\cos n^{\circ }}\end{array}$

Hence proved.