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

Question 810211: Determine sin(2x)
when cos(x)=-1/3 and sin(x) is negative
Answer by DrBeeee(684) (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 to get
(4) or
(5) or
(6) or
(7)
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) or
(10)
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:

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