\( \tan x = \frac { 3 } { 4 } , \cos x < 0 \)

We are given that tan x = 3/4 and that cos x < 0. Step 1. Identify the quadrant. • Since tan x = sin x/cos x is positive, sin x and cos x must have the same sign. • Given that cos x is negative, it follows that sin x must also be negative. • Therefore, x lies in the third quadrant. Step 2. Find sin x and cos x. Imagine a right triangle where the opposite side = 3 and the adjacent side = 4 (ignoring signs initially). Then, the hypotenuse is   hypotenuse = √(3² + 4²) = √(9 + 16) = √25 = 5. In the standard triangle (first quadrant), one would have:   sin x = 3/5 and cos x = 4/5. However, since x is in the third quadrant, both sine and cosine must be negative. Therefore,   sin x = –3/5 and cos x = –4/5. Step 3. Verify: • tan x = sin x/cos x = (–3/5)/(–4/5) = 3/4, as given. • cos x < 0, as required. Thus, the values satisfy the conditions. Final Answer: sin x = –3/5, cos x = –4/5, and tan x = 3/4. Additionally, one can express all solutions for x as:   x = π + arctan(3/4) + πk (for any integer k)