SOLUTION: Determine sin(2x) when cos(x)=-1/3 and sin(x) is negative

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Question 810211: Determine sin(2x)
when cos(x)=-1/3 and sin(x) is negative
Answer by DrBeeee(684) About Me  (Show Source):
You can put this solution on YOUR website!
Use the double angle formula
(1) sin(2*x) = 2*sin(x)*cos(x)
and the givens
(2) cos(x) = -1/3 and
(3) sin(x) < 0
Use sin%5E2%28x%29+%2B+cos%5E2%28x%29+=+1 to get
(4) sin%5E2%28x%29+%2B+%28-1%2F3%29%5E2+=+1 or
(5) sin%5E2%28x%29++=+1+-1%2F9 or
(6) sin%5E2%28x%29+=+8%2F9 or
(7) sin%28x%29+=+-+sqrt%288%29%2F3
where we select the negative value in (7) because of the given condition (3).
Now put sin(x) and cos(x) into (1) to get
(8) sin(2*x) = 2*(- sqrt(8)/3)*(-1/3) or
(9)+sin%282%2Ax%29+=+2%2Asqrt%288%29%2F9 or
(10) sin%282%2Ax%29+=+4%2Asqrt%282%29%2F9
Let's use numerical values to check the answer.
From (2) we get
(11) x = arccos(-1/3) or
(12) x = 109.47+
Then we get
(13) sin(x) = -0.9428
and
(14) sin(2x) = sin(218.94) or
(15) sin(2x) = 0.6285
Now
Is (sin(2*x) = 2*sin(x)*cos(x))?
Is (0.6285 = 2*(-09428)*(-1/3))?
Is (0.6285 = 0.6285)? Yes
Answer:sin%282%2Ax%29+=+4%2Asqrt%282%29%2F9