trig functions?

Question

find the exact value of each of the remaining trig functions of theta.

sin theta = -2/3, 180 degrees < theta < 270 degrees

how do I do this?? Please help and explain, thanks

Answer 1

sin^2 theta + cos^2 theta = 1 means |cos theta| = sqrt (1-sin^2 theta) = sqrt (1-4/9) = sqrt(5)/3 (absolute value).

we know that theta is between 180 and 270 which means |cos theta| = -cos theta.
cos theta = -sqrt(5)/3

tan theta = sin theta/cos theta = (-2/3)/(-sqrt(5)/3)) = 2/sqrt(5)

you can also use this method


sin theta = -2/3 and theta is between 180 and 270 means that we're dealing with a lower triangle with an hypotenuse of 3 and an opposite side of 2. then the adjacent side is sqrt(3^2 - 2^2) = sqrt(9-4) = sqrt(5) (Pythagoras's theorem). We know that cos = adjacent side/hypotenuse. then in this case:
sqrt(5)/3. But as it's a lower triangle, cos theta is -sqrt(5)/3.

tan = opposite/adjacent = 2/sqrt(5)

in both methods:

cot theta = 1/tan theta = sqrt(5)/2
sec theta = 1/cos theta = -3/sqrt(5)
csc theta = 1/ sin theta = -3/2

Answer 2

since theta is in between 180 and 270 degrees , it is in third quadrant.
in third quadrant except tan and cot all others are negative
sin theta = - 2/3
ignore negative sign
in right angled triangle
sin theta = oppsite side of theta/hypotenuse = 2/3
so square of adjacent side to theta = 3^2 - 2^2 = 9 - 4 = 5
adjacent side = sqrt(5)
cos theta = sqrt(5)/3 = - sqrt(5)/3( since cos is negative in third quadrant)
sec theta = 1/cos theta = -3/sqrt(5)
csc theta = 1/ sin theta = - 3/2
tan theta = sin theta/cos theta = (2/3)3/sqrt(5) = 2/sqrt(5)
cot theta = 1/tan theta = sqrt(5)/2

Answer 3

I am assuming that you can use a calculator to do this.

type 2 / 3, then find the sin^(-1) [arcsin] fn on your calc

Mine gave me 41.81 deg

Now when theta is between 180 and 270, so add 180 to 41.8
= 221,81