If tan(θ) = 15/8 and θ is in quadrant 3, we can use the Pythagorean identity to find the other trigonometric ratios.
We know that tan(θ) = opposite/adjacent, so we can use the Pythagorean theorem to find the hypotenuse. Let's assume that the opposite side is 15 and the adjacent side is 8. Then, using the Pythagorean theorem, we can find the hypotenuse:
h^2 = 15^2 + 8^2
h^2 = 225 + 64
h^2 = 289
h = √289
h = 17
Now that we know all three sides of the triangle, we can find the other trigonometric ratios:
sin(θ) = opposite/hypotenuse = 15/17
cos(θ) = adjacent/hypotenuse = -8/17 (since it's in quadrant 3, the x-coordinate is negative)
csc(θ) = 1/sin(θ) = 17/15
sec(θ) = 1/cos(θ) = -17/8 (since it's in quadrant 3, the x-coordinate is negative)
cot(θ) = 1/tan(θ) = 8/15
So, the other trigonometric ratios for θ in quadrant 3 are:
sin(θ) = 15/17
cos(θ) = -8/17
csc(θ) = 17/15
sec(θ) = -17/8
cot(θ) = 8/15
