Given r, x, y, θ of a triangle, solve for the missing parts.

a. r = 13, x =–5;
b. x = 7, y = –11;
c. θ = 133˚ and r = 44.

To solve for the missing parts in the triangle using the given information, we can refer to the trigonometric relationships and formulas for triangles.

a. Given r = 13 and x = -5, we need to find y and θ.

To find y, we can use the Pythagorean theorem:
y = √(r^2 - x^2)
y = √(13^2 - (-5)^2)
y = √(169 - 25)
y = √144
y = 12

To find θ, we can use the inverse tangent (arctan) function:
θ = arctan(y / x)
θ = arctan(12 / -5)
θ ≈ -67.38°

b. Given x = 7 and y = -11, we need to find r and θ.

To find r, we can again use the Pythagorean theorem:
r = √(x^2 + y^2)
r = √(7^2 + (-11)^2)
r = √(49 + 121)
r = √170
r ≈ 13.04

To find θ, we can use the inverse tangent function:
θ = arctan(y / x)
θ = arctan(-11 / 7)
θ ≈ -57.99°

c. Given θ = 133˚ and r = 44, we need to find x and y.

To find x, we can use the cosine function:
x = r * cos(θ)
x = 44 * cos(133°)
x ≈ -14.75

To find y, we can use the sine function:
y = r * sin(θ)
y = 44 * sin(133°)
y ≈ 37.18

So, for part a, the missing parts are y = 12 and θ ≈ -67.38°.
For part b, the missing parts are r ≈ 13.04 and θ ≈ -57.99°.
For part c, the missing parts are x ≈ -14.75 and y ≈ 37.18.