Respuesta :
let's recall that on the II Quadrant x/cosine is negative whilst y/sine is positive,
also let's recall that the hypotenuse is simply the radius distance and thus is never negative.
[tex]\bf cot(\theta )=\cfrac{\stackrel{adjacent}{-2}}{\stackrel{opposite}{1}}\impliedby \textit{let's find the hypotenuse} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=\sqrt{a^2+b^2} \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ c=\sqrt{(-2)^2+1^2}\implies c=\sqrt{5} \\\\[-0.35em] ~\dotfill\\\\ csc(\theta )=\cfrac{\stackrel{hypotenuse}{\sqrt{5}}}{\stackrel{opposite}{1}}\implies csc(\theta )=\sqrt{5}[/tex]
The exact value of csc theta when the value of cot theta is -2 and the terminal side of theta lies in quadrant II (2) is √(5).
What is the terminal side of an angle?
The terminal side of an angle is the rotated side of the initial side around a point to form an angle. This rotation can be clockwise or counter clock wise.
The exact value of csc theta has to be found out. The value of cot theta is,
cot θ = -2
cot θ =-2/1.
By the property of right angle triangle, the ratio of adjacent side to the opposite side is equal to the cot theta. Thus,
Adjacent side= -2
Opposite side= 1
The value of hypotenuse side is equal to the square root of the sum of the square of adjacent side and opposite side. Thus,
Hypotenuse side=√((-2)²+1²)
Hypotenuse side=√(5)
By the property of right angle triangle, the ratio of hypotenuse side to the opposite side is equal to the coses theta. Thus,
coses θ =√(5)/1
coses θ =√(5)
Hence, the exact value of csc theta when the value of cot theta is -2 and the terminal side of theta lies in quadrant II (2) is √(5).
Learn more about the terminal side of an angle here
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