SOLUTION: If theta is a third quadrant angle and cos theta = -2/3, find sin theta.

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Question 1138396: If theta is a third quadrant angle and cos theta = -2/3, find sin theta.
Found 3 solutions by josmiceli, MathLover1, ikleyn:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
The sin is also negative in the 3rd quadrant
Find vertical coordinate
+b%5E2+=+3%5E2+-+%28-2%29%5E2+
+b%5E2+=+9+-+4+
+b%5E2+=+5+
+b+=+-sqrt%285%29+
------------------------
+sin%28+theta+%29+=+-sqrt%285%29%2F3+

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

If theta is a third quadrant angle and cos%28+theta%29+=+-2%2F3, find sin+%28theta%29.
From the sign on the cosine value, I only know that the angle is in QII or QIII. That's why they had to give me that additional specification: so I'd know which of those quadrants I'm in.

cos%28+theta%29+=+-2%2F3=>x%2Fr
The Pythagorean Theorem gives me:
r%5E2+=+x%5E2+%2B+y%5E2+
3%5E2+=+%28-2%29%5E2+%2B+y%5E2+
9+=+4+%2B+y%5E2+
9-4+=+y%5E2+
y=±sqrt%285%29
Since in QIII, below the x-axis, so y is negative.
I'll take the negative solution to the equation, and y=+-sqrt%285%29
sin+%28theta%29=y%2Fr
sin+%28theta%29=-sqrt%285%29%2F3

Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.

sin%28theta%29 = -sqrt%281+-+cos%5E2%28theta%29%29 = -sqrt%281+-+%28-2%2F3%29%5E2%29 = -sqrt%281+-+4%2F9%29 = -sqrt%285%2F9%29 = -sqrt%285%29%2F3.


The sign at the square root  is " - " (MINUS), because the angle theta is in QIII, by the condition.


It is your standard  "mantra",  which you should pronounce every time,  when you solve such a problem.