Question

Find the exact value of the expression:
csc(cos^(-1)(-1/3))

Answer 1

\[\csc(\cos ^{-1}(\frac{ 1 }{ 3 }))\]

Answer 2

suppose instead you were asked, "if \(\cos(x)=\frac{1}{3}\), what is \(\sin(x)\)? "
could you solve it?

Answer 3

1/3 isn't on the unit circle

Answer 4

yes, \(\frac{1}{3}\) is some first coordinate on the unit circle, because it is between -1 and 1

Answer 5

there is a picture of an angle whose cosine is \(\frac{1}{3}\)
if you want the sine of that angle you need to solve for the opposite side using pythagoras
it is \(\sqrt{3^2-1^2}=\sqrt{8}=2\sqrt{2}\)

Answer 6

so the sine of your angle is \(\frac{2\sqrt{2}}{3}\) and therefore the cosecant is \(\frac{3}{2\sqrt{2}}\)

Answer 7

Isn't csc 1/sin?

Answer 8

aka the reciprocal