Question

Find the value of the trigonometric expression: $\csc \left( {\dfrac{{5\pi }}{4}} \right)$

Answer 1

Hint:We are given a trigonometric function of cosecant with an angle of $\dfrac{{5\pi }}{4}$ and we have to find its value. For this we use the Cartesian plane for trigonometric functions’ sine i.e. positive or negative and use the standard table of trigonometric functions value at particular angles.

Complete step by step solution:
Firstly we write down the expression of which we are supposed to find the value
$\csc \left( {\dfrac{{5\pi }}{4}} \right)$
We know that the trigonometric function is constant here also the angles provided are in radian.
So in our first step we try to change the radian angle into degree angle to get a better understanding of where it lies in the Cartesian plane. We have the conversion of radians into degrees like this:
${\text{Radians}} \times \dfrac{{180^\circ }}{\pi } = {\text{Degree}}$
So using this formula of conversion we convert our argument of the function (angle in this case) like this:
\[\dfrac{{5{\pi }}}{4} \times \dfrac{{180^\circ }}{{{\pi }}} = \dfrac{{5 \times
{{\mathop {180}\limits}}}}{{{4}}} = 5 \times 45 = 225^\circ \]
This means our angle is $225$ degrees. Now we find the value of $\csc 225^\circ $ we can say by looking at the angle that this cosecant will lie in the third quadrant of the Cartesian plane. The below figure will give a better idea of this fact

By careful observation of figure (a) we can say this that the conversion would happen in the third quadrant where cosecant is negative and thus we have
$\csc (180^\circ + 45^\circ ) = - \csc 45^\circ $
So we have got the simplified of our given expression as $ - \csc 45^\circ $ now looking this value up in the below table we have

Using the table we have
$ - \csc 45 = - \sqrt 2 $
This is our final value of the given cosecant function.

Note: Wherever possible conversion of trigonometric functions should be done with respect to either $180^\circ $ or$0^\circ ,360^\circ $for simpler calculations. Because this conversion does not lead to conversion of function such as sine converts into cosine which happens while doing so with $90^\circ \,{\text{or}}\,270^\circ$