Given `sin(theta)=(-7)/8` , find `1/(cot(theta))` in the interval `(3pi)/2<theta<2pi` .
(1) The interval indicates we are to work in the fourth quadrant.
(2) Draw a triangle in the fourth quadrant with a vertex at the origin. The angle `theta` will be measured off of the positive x-axis.
Since `sin(theta)` is the ratio of the side opposite `theta` over the hypotenuse, we can use the pythagorean theorem to find the third side of the triangle (and thus the x-coordinate of the remaining points on the triangle). The third side: `a^2+(-7)^2=8^2=>a^2=15=>a=sqrt(15)`
Thus the vertices of the triangle are at (0,0),`(sqrt(15),0),(sqrt(15),-7)`
(3) `1/(cot(theta))=tan(theta)` . Since `tan(theta)` is the ratio of the side opposite `theta` to the side adjacent to `theta` we have `tan(theta)=(-7)/(sqrt(15))` . Rationalizing we get `1/(cot(theta))=tan(theta)=(-7)/(sqrt(15))=(-7sqrt(15))/15`