Question

Find the exact value of s in the given interval that has the given circular function value.

Answer 1

Recall that:

`\tan x=(\sin x)/(\cos x)\text{.}`

Therefore:

`\tan s=1\Leftrightarrow(\sin s)/(\cos s)=1.`

Then:

`\sin s=\cos s\text{.}`

Now, notice that:

`\sin s-\cos s=-√(2)\cos (s+(\pi)/(4)).`

Then:

`-\sqrt[]{2}\cos (s+(\pi)/(4))=0.`

Therefore:

`\cos (s+(\pi)/(4))=0.`

Then:

`\begin{gathered} s+(\pi)/(4)=(\pi)/(2)+n\pi, \\ s+(\pi)/(4)=(3\pi)/(2)+n\pi\text{.} \end{gathered}`

Therefore:

`\begin{gathered} s=(\pi)/(4)+n\pi, \\ s=(5\pi)/(4)+n\pi\text{.} \end{gathered}`

Since:

`s\in\lbrack\pi,(3\pi)/(2)\rbrack^{},`

we get that:

`s=(5\pi)/(4)\text{.}`

Answer:

`s=(5\pi)/(4)\text{.}`