Find sin theta and tan theta if cos theta equals .8 and tan theta is less than 0

Since  cos(θ) = adjacent / hypotenuse  in a right triangle and

since cos(θ) = 0.8  =  8/10   --->   adjacent = 8  and  hypotenuse  =  10.

Using the Pythagorean Theorem, opposite = +6  or  -6, depending upon the quadrant.

Since tan(θ) < 0, the triangle is found in either the second quadrant or the fourth quadrant.

Since cos(θ) > 0, the triangle is found in either the first quadrant or the fourth quadrant.

Therefore, it must be in the fourth quadrant.

In the fourth quadrant, sin(θ) < 0, making the opposite side -6.

--->   sin(θ)  =  -6/10  =  -0.60.

--->   tan(θ)  =  -6/8  =  -0.75.