Chapter 12.2, Problem 19WE

To determine

Use the cofunction identities to find the measure of the acute angle $\theta$ .

  $\sec \varphi =\csc 80°$

Answer to Problem 19WE

  $\varphi =10°$

Explanation of Solution

Given:

  $\sec \varphi =\csc 80°$

Formula Used:

  $\csc \theta =\sec \left(90°-\theta \right)$

Calculation:

  $\begin{array}{l}\sec \varphi =\csc 80°\\ \Rightarrow \sec \varphi =\sec \left(90°-80°\right)\mathrm{...........}[\csc \theta =\sec \left(90°-\theta \right)]\\ \Rightarrow \sec \varphi =\sec 10°\\ \Rightarrow \varphi =10°\mathrm{.................................}[\sec x=\sec y\iff x=y\text{ in first quadrant]}\end{array}$