Question

Given sin theta = 2/3 and pi/2<theta<pi how do you find the value of other 5 other trigonometric functions?

Answer 1

`sin(theta) = 2/3color(white)("XXXXXXX")csc(theta)=3/2`
`cos(theta) = -sqrt(5)/3color(white)("XXXX")sec(theta)=-3/sqrt(5)`
`tan(theta)=-2/sqrt(5)color(white)("XXXX")cot(theta)=-sqrt(5)/2`

Explanation:

`pi/2 < theta < pi`
tells us that `theta` is in Quadrant II

`sin(theta)=2/3`
tells us that the side ratios are
`color(white)("XXX")`opposite side: `2`
`color(white)("XXX")`hypotenuse: `3`
and using the Pythagorean Theorem and the fact that we are in Q II
`color(white)("XXX")`adjacent side: `-sqrt(5)`

By definition:
`sin = ("opposite")/("hypotenuse")color(white)("XXXX")csc=("hypotenuse")/("opposite")`

`cos = ("adjacent")/("hypotenuse")color(white)("XXXX")sec=("hypotenuse")/("adjacent")`

`tan = ("opposite")/("adjacent")color(white)("XXXX")cot=("adjacent")/("opposite")`