I have to keep pointing out that most trig only uses two triangles. This is another 45/45/90 and 30/60/90 problem.
A line through rectangular coordinates (x_1, y_1)(x1,y1) and (x_2,y_2)(x2,y2) looks like
(y - y_1)(x_2 - x_1) = (x - x_1)(y_2 - y_1)(y−y1)(x2−x1)=(x−x1)(y2−y1)
For polar coordinates
x = r cos thetax=rcosθ
y = r sin thetay=rsinθ
x_1 = 3 cos(pi/4) = 3/2 sqrt 2x1=3cos(π4)=32√2
y_1 = 3 sin( pi/4) = 3/2 sqrt 2y1=3sin(π4)=32√2
x_2 = 2 cos ({7pi}/6) = -sqrt{3}/2 x2=2cos(7π6)=−√32
y_2 = 2 sin({7 pi}/6)=-1/2y2=2sin(7π6)=−12
( r sin theta - 3/2 sqrt 2)(-sqrt{3}/2 - 3/2 sqrt 2) = (r cos theta- 3/2 sqrt 2)(-1/2 - 3/2 sqrt 2)(rsinθ−32√2)(−√32−32√2)=(rcosθ−32√2)(−12−32√2)
We could stop here but let's clear the fractions and a minus sign.
(2 r sin theta - 3 sqrt 2)(sqrt{3} + 3 sqrt 2) = (2 r cos theta- 3 sqrt 2)(1 + 3 sqrt 2) (2rsinθ−3√2)(√3+3√2)=(2rcosθ−3√2)(1+3√2)
Check: Alpha
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