Question

If cot x = 8 & csc < 0 Find sin x & cos x

Answer 1

well, csc(x) < 0, is just another way of saying csc(x) is negative, now, we know that csc(x) is the reciprocal of sin(x), so if the cosecant is negative, the sine is also negative.

`\cot(x)=8\implies \cot(\theta )=\cfrac{\stackrel{adjacent}{8}}{\underset{opposite}{1}}\hspace{5em} \textit{let's find the \underline{hypotenuse}} \\\\\\ \begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2\implies c=√(a^2 + o^2) \end{array} \qquad \begin{cases} c=hypotenuse\\ a=\stackrel{adjacent}{8}\\ o=\stackrel{opposite}{1} \end{cases} \\\\\\ c=√( 8^2 + 1^2)\implies c=√( 64 + 1 ) \implies c=√( 65 ) \\\\[-0.35em] ~\dotfill`

`\cos(x )=\cfrac{\stackrel{adjacent}{8}}{\underset{hypotenuse}{√(65)}}\implies \cos(x )=\cfrac{8√(65)}{65} \\\\\\ \sin(x )=\cfrac{\stackrel{opposite}{-1}}{\underset{hypotenuse}{√(65)}}\implies \sin(x )=-\cfrac{√(65)}{65}`