Do the rectangular coordinatesand the polar coordinatesrepresent the same(− 8,− 15)(17,− 118) point? Mathematically justify your response and explain your reasoning.

Answer 1

(-8,-15) = (x,y)

x and y are sides of a right triangle, with hypotenuse h

such that h^2 = 8^2 +15^2 = 64+225 = 289

h= sqr289 = 17 = r = radius for the polar coordinate

now just need the angle = 180 + arctan(-15/-8) = arctan(15/8) = about 180+61.9275 = about 241.9275

= about 242 degrees

polar coordinates = about (17, 242) not (17, -118)

Answer 2

Let's convert the rectangular coordinates: (-8, -15) into polar coordinates or (r, θ)

To find r, use the Pythagorean Theorem:

r = √(8² + 15²) = 17

To find θ, using inverse tangent:

θ = tan^-1(8/15) = 28° but this is in Q3 so one way to get there is go CW by 90° and then the other 28° which makes -118°

So the polar coordinates of the same point are (17, -118°)