Find the coordinates of the point on a circle centered at the origin with radius 9 corresponding to an angle of 135∘.

Question

This seems fairly simple but I am thinking about it too hard, I am trying to learn the process.

Answer 1

x^2 + y^2 = 9^2

135 degrees puts the point on the circle in the 2nd quadrant with x<0 and y>0 but the absolute value of x - y

x=-y

y^2 + y^2 = 81

2y^2 = 81

y^2 = 81/2

y = 9/sqr2 = and x=-9/sqr2

y= about 6.364 and x =about -6.364

(x,y) = (-9/sqr2, 9/sqr2) or (-9sqr2/2, 9sqr2/2) = approximately (-6.4, 6.4)

x= rcos135 = 9(-.707) = -6.363 y=rsin135 = 9(.707) = 6.363

converting from polar to rectangular coordinates, use x=rcos(thet) and y=rsin(theta)

and x^2 + y^2 = r^2

Answer 2

sin 135° = sin 45°

cos 135° = - cos 45°

sin 45° = √2 / 2, y / 9 = √2 / 2, y = 9√2 / 2

-cos 45° = -√2 / 2, x / 9 = -√2 / 2, x = -9√2 / 2