Leave answers in simplified form. Convert from polar to rectangular coordinates.

Question

Convert each of the following as indicated.

a) Convert from polar to rectangular coordinates:

(7,𝜋/6)

b.) Convert from rectangular to polar coordinates:

−1+√3𝑖

I need help with these problems, please!

Answer 1

a)

(r,θ)=(7,π/6)

rcos(θ)=7cos(π/6)=7√3/2

rsin(θ)=7sin(π/6) = 7/2

(x,y) = (7√3/2, 7/2)

b)

a+bi = -1 + i√3

r = √(a²+b²) = √4 = 2

θ = tan^-1(b/a) = tan^-1(√3/ -1) = 2π/3

(r,θ)=(2,2π/3)

Answer 2

pi/6 = 30 degrees. pi=180 degrees. 180/6 = 30

7 is the radius, the hypotenuse of a right triangle, where the opposite side = half the hypotenuse: 7/2 = 3.5

and the adjacent side is sqr3 times 7/2 = 3,5 sqr3 = about 3.5(1.732)

rectangular coordinates are (x,y) where x = 3.5sqr3 and y= 3.5

(x,y) = (3.5sqr3, 3.5)

-1 + sqr3 i is in 2nd quadrant. x coordinate is -1, y coordinate is isqr3. the angle formed has a reference angle of 60 degrees. Actual angle is 180-60 = 120 degrees = 2pi/3 radians

polar coordinates are (2, 2pi/3)

sqr3 and 1 are sides of a right triangle with hypotenuse = 2