Answer:
[tex]-\frac{3}{5}[/tex]
Step-by-step explanation:
Remember that the tangent trigonometric ratio is the opposite side of right triangle divided by the adjacent side:
[tex]tan(\alpha )=\frac{opposite-side}{adjacent-side}[/tex]
[tex]tan(\alpha )=-\frac{4}{3}[/tex]
Comparing the equations we can infer that:
opposite side = 4
adjacent side = 3
Now we can use Pythagoras to find the hypotenuse of our right triangle:
[tex]hypotenuse^2=side^2+side^2[/tex]
[tex]hypotenuse^2=4^2+3^2[/tex]
[tex]hypotenuse^2=25[/tex]
[tex]hypotenuse=\sqrt{25}[/tex]
[tex]hypotenuse=5[/tex]
Remember that the cosine trigonometric ratio is the adjacent side divided by the hypotenuse; in other words:
[tex]cos(\alpha)=\frac{adjacent-side}{hypotenuse}[/tex]
We know that adjacent side = 3 and hypotenuse = 5.
Replacing values:
[tex]cos(\alpha )=\frac{3}{5}[/tex]
Now, remember that cosine means x and sine means y. In Quadrant 2 x is negative, which means that cosine is negative.
So, if [tex]tan(\alpha )=-\frac{4}{3}[/tex] in quadrant 2, then [tex]cos(\alpha )=-\frac{3}{5}[/tex]
