SOLUTION: if tan(x)=(-4/3), cos(x)*cot(x)<0, what is the value of ((4cos(x)+1)/(3sin(x)+5))? the book says (17/13), but I keep getting (-13/17).

Question 369432: if tan(x)=(-4/3), cos(x)*cot(x)<0, what is the value of ((4cos(x)+1)/(3sin(x)+5))?
the book says (17/13), but I keep getting (-13/17).
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

It may help to draw a right triangle and pick one of the acute angles to be x. Since tan is the ratio of opposite over adjacent, there are two ways for this ratio to work out to -4/3:

If the opposite side is -4 and the adjacent side is 3, or
If the opposite side is 4 and the adjacent side is -3

cos(x)*cot(x)<0
If tan is negative, so will cot. So for the product of cos and cot to be negative, cos must be positive. Since cos is adjacent over hypotenuse and since it must be positive, the adjacent side must be the positive one. Now we know which of the possibilities described above is correct: opposite side is -4 and adjacent side is 3.

For ((4cos(x)+1)/(3sin(x)+5)) we need cos(x) and sin(x). For both of these we need the hypotenuse. We can use the Pythagorean Theorem to find the hypotenuse (which we'll call h):

(since we never use a negative hypotenuse).
So sin(x) = opposite/hypotenuse = (-4)/5 and
cos(x) = adjacent/hypotenuse = 3/5
With these we can find
((4cos(x)+1)/(3sin(x)+5))

which simplifies as follows:

Multiplying the numerator and denominator by 5 we get:

So the book is correct.
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