Question

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

Answer 1

Step 1:

Given data

`\theta\text{ = }(9\pi)/(4)`

1.)

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

2.)

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

3.)

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

4.)

`\begin{gathered} cot(9\pi)/(4)\text{ = }(1)/(tan(9\pi)/(4))\text{ = }(1)/(1)\text{ = 1} \\ \\ cot\theta\text{ = 1} \end{gathered}`

Final answer