Are ${csc θ}^{2}$ $csctheta^2$ and ${csc}^{2} θ$ $csc^2theta$ same or different?

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Are ${csc θ}^{2}$ $csctheta^2$ and ${csc}^{2} θ$ $csc^2theta$ same or different?

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Answer 1 · 2017-02-12T14:35:40

${csc θ}^{2}$ $csctheta^2$ and ${csc}^{2} θ$ $csc^2theta$ are different. Please see below for details.

Explanation:

${csc θ}^{2}$ $csctheta^2$ and ${csc}^{2} θ$ $csc^2theta$ are different.

Let us see the figure below.
enter image source here
From the definition of trigonometric ratios, $csc θ = \frac{hypotenuse}{opposite side}$ $csctheta=("hypotenuse")/("opposite side")$

and ${csc}^{2} θ = {(csc θ)}^{2} = csc θ \times csc θ$ $csc^2theta=(csctheta)^2=cscthetaxxcsctheta$

= $\frac{{(hypotenuse)}^{2}}{{(opposite side)}^{2}}$ $("hypotenuse")^2/("opposite side")^2$

However, ${csc θ}^{2}$ $csctheta^2$ is the cosecant ratio of $θ^{2}$ $theta^2$ .

Now if $θ$ $theta$ is in degrees, $θ^{2}$ $theta^2$ does not work out.

Hence while considering $θ^{2}$ $theta^2$ , we consider it only in radians.

Now for example , if $θ = \frac{π}{3}$ $theta=pi/3$

${csc}^{2} θ = {csc}^{2} (\frac{π}{3}) = {(\frac{2}{\sqrt{3}})}^{2} = \frac{4}{9} = 0.4444$ $csc^2theta=csc^2(pi/3)=(2/sqrt3)^2=4/9=0.4444$

For working out ${csc θ}^{2}$ $csctheta^2$ , we have

$θ = \frac{π^{2}}{9} = \frac{9.8696}{9} = 1.0966$ $theta=pi^2/9=9.8696/9=1.0966$

and using scientific calculator, with $θ = 1.0966$ $theta=1.0966$ in radians

${csc θ}^{2} = csc 1.0966 = 1.124$ $csctheta^2=csc1.0966=1.124$

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Are ${csc θ}^{2}$ $csctheta^2$ and ${csc}^{2} θ$ $csc^2theta$ same or different?

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Explanation:

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