Find cos theta if sin theta = 2/3. assume the terminal side of the angel falls in quadrant 2

Well knowing that the terminal arm of the standard position angle is in quadrant 2, we can determine the reference angle, in quadrant 2, by simply taking the difference between 180 and whatever the angle is.

So ø reference = 180 - ø in standard position.

Regardless, the reference angle is in quadrant 2, we need to then label the sides of the reference triangle based on the opposite and hypotenuse.

Solve for adjacent side using Pythagoras theorem.

A^2 = C^2 - B^2

A^2 = 3^2 - 2^2

A^2 = 9 - 4

A^2 = 5

A = sq root of 5.

Then write the cos ratio using the new side.

Cos ø = ✔️5/3. Place a negative in front of cos ø as cos is negative in second quadrant.