To solve this, first, we must determine x. We are given that tangent of x is -40/9 and that x is in the fourth quadrant. The ratio of tangent is opposite/adjacent. Thus, in this case, the opposite side is -40 and the adjacent side is 9.
Since tangent is negative in the fourth quadrant, we can say that the reference angle is atan(40/9). We will also need to keep in mind that the angle x is between 270° and 360°, or between 3π/2 and 2π in radians, because it is in the fourth quadrant.
Now, let's find the values of sin(x), cos(x), and tan(x).
We have the formula cos²(x) + sin²(x) = 1, which can be rearranged to find either sin(x) or cos(x) given that the other one is known.
1. sin(x) can be found by rearranging to sin(x) = sqrt(1 - cos²(x)).
Here, cos(x) can be found using the formula for cos(x) in terms of tan(x) which is cos(x) = sqrt(1 /(1 + tan²(x))), by substituting tan(x) = -40/9, we get cos(x) = sqrt(1/(1 + (−40/9)²)).
So, substituting cos(x) into the formula for sin(x), we get sin(x) = sqrt(1 - 1/(1 + (−40/9)²)).
2. To find cos(x), substitute tan(x) into the formula for cos(x).
Thus, cos(x) = sqrt(1 /(1+(-40/9)²)).
3. tan(x) can be determined directly from the aforementioned information.
tan(x) = -40/9.
In conclusion, the exact values of sin(x), cos(x) and tan(x) can be determined using the formulas and information given above, with x existing in the fourth quadrant and having a tangent of -40/9.