Given: cosθ = - 4/5, sin x = - 12/13, θ is in the third quadrant, and x is in the fourth quadrant; evaluate tan (θ + x) and sin (θ + x))

tan (θ+x) = - 33/56

sin (θ+x) = 33/65

Step-by-step explanation:

cosθ = - 4/5, sin x = - 12/13

cosθ = - 4/5; θ is in the third quadrant=> sinθ=-√ (1-cos² θ)

sinθ = - √ (1-16/25) = - √ (9/25) = -3/5

sinx = - 12/13; x is in the fourth quadrant=> cosx=+√ (1-sin²x)

cosx=√ (1-144/169) = √ (25/169) = 5/13

tgθ = sinθ/cosθ = (-3/5) / (-4/5) = 3/4

tgx = sinx/cosx = (-12/13) / (5/13) = - 12/5

tan (θ+x) = (tgθ+tgx) / (1-tgθ*tgx)

tan (θ+x) = (3/4-12/5) / (1+3/4*12/5) = (-33/20) / 56/20) = - 33/56

sin (θ+x) = sinθcosx+coxθsinx=-3/5 * 5/13 + -4/5 * (-12/13) = -15/65+48/65=33/65