What is the exact value of cosØ when Ø lies in quadrant 1 and sinØ= the square root of 2 over 6?

Let cos(Ø) = x/r.

Let sin(Ø) = y/r

Since sin(Ø) = sqrt(2) / 6     --->     y = sqrt(2)     and     r = 6.

By the Pythagorean Identity,  x² + y²  =  r²     --->     x² + ( sqrt(2) )²  =  6²     --->     x² + 2  =  34     --->     x  =  sqrrt(34)

Therefore  cos(Ø)  =  sqrt(34) / 6