find sin s

when cot s= -1/3, s in quadrant 4

If cot s= -1/3

then tan s = -3/1

construct a triangle, by Pythagoras, the hypotenuse is √10 , and in the 4th quad, the sine is negative, so
sin s = -3/√10

To find the value of sin(s), we will use the given information that cot(s) = -1/3 and that s is in quadrant 4.

In quadrant 4, the cosine and cotangent are positive.

First, we need to find the value of cot(s) in quadrant 1 since cotangent is positive in that quadrant.
To do this, we can use the reciprocal property of cotangent:
cot(s) = -1/3 => tan(s) = -3/1 => sin(s) / cos(s) = -3/1.

Since the cosine is positive in quadrant 1, we know that cos(s) = sqrt(1 - sin^2(s)), where sin(s) is the value we are trying to find.

We can substitute this value into the equation above:
sin(s) / sqrt(1 - sin^2(s)) = -3/1.

Now, to solve for sin(s), we will cross-multiply and square both sides of the equation:

sin^2(s) = (-3/1)^2 * (1 - sin^2(s)).
sin^2(s) = 9 * (1 - sin^2(s)).
sin^2(s) = 9 - 9sin^2(s).
10sin^2(s) = 9.
sin^2(s) = 9/10.

Finally, taking the square root of both sides, we can solve for sin(s):

sin(s) = sqrt(9/10).
sin(s) = 3/sqrt(10).
sin(s) = 3√10/10.

Therefore, in quadrant 4 where s is located, sin(s) = 3√10/10.