Question

If theta= 9 pi 4 then find exact values for the following

Answer 1

Solution

Step 1:

Given data

$\theta\text{ = }\frac{9\pi}{4}$

1.)

$\begin{gathered} \sec \left(\frac{9\pi }{4}\right) \\ \mathrm{Express\:with\:sin,\:cos} \\ \\ \sec \left(\frac{9\pi }{4}\right)=\frac{1}{\cos \left(\frac{9\pi }{4}\right)} \\ \\ =\frac{1}{\cos \left(\frac{9\pi }{4}\right)} \\ \\ =\frac{1}{\cos \left(\frac{\pi }{4}\right)} \\ \\ \mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(\frac{\pi }{4}\right)=\frac{\sqrt{2}}{2} \\ \\ =\frac{1}{\frac{\sqrt{2}}{2}} \\ \\ sec\theta\text{ = }\sqrt{2} \end{gathered}$

2.)

$\begin{gathered} \csc \left(\frac{9\pi }{4}\right) \\ \\ \csc \left(\frac{9\pi }{4}\right)=\frac{1}{\sin \left(\frac{9\pi }{4}\right)} \\ \\ =\frac{1}{\sin \left(\frac{\pi }{4}\right)} \\ \\ \mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(\frac{\pi }{4}\right)=\frac{\sqrt{2}}{2} \\ \\ =\frac{1}{\frac{\sqrt{2}}{2}} \\ \\ csc\theta\text{ = }\sqrt{2} \end{gathered}$

3.)

$\begin{gathered} \tan \left(\frac{9\pi }{4}\right) \\ \\ =\tan \left(\frac{\pi }{4}\right) \\ \\ \mathrm{Use\:the\:following\:trivial\:identity}:\quad \tan \left(\frac{\pi }{4}\right)=1 \\ tan\theta\text{ =1} \end{gathered}$

4.)

$\begin{gathered} cot\frac{9\pi}{4}\text{ = }\frac{1}{tan\frac{9\pi}{4}}\text{ = }\frac{1}{1}\text{ = 1} \\ \\ cot\theta\text{ = 1} \end{gathered}$

Final answer