Trigonometric Value Calculation
We are given that $\cos x = -\frac{3}{5}$ and the angle $x$ lies in the third quadrant.
We need to find the value of $\sin x$.
Using Trigonometric Identity
Recall the fundamental trigonometric identity:
$ \sin^2 x + \cos^2 x = 1 $
Substitute the given value of $\cos x$ into the identity:
$ \sin^2 x + \left(-\frac{3}{5}\right)^2 = 1 $
$ \sin^2 x + \frac{9}{25} = 1 $
Now, solve for $\sin^2 x$:
$ \sin^2 x = 1 - \frac{9}{25} $
$ \sin^2 x = \frac{25 - 9}{25} $
$ \sin^2 x = \frac{16}{25} $
Determining the Sign
Taking the square root of both sides gives:
$ \sin x = \pm \sqrt{\frac{16}{25}} $
$ \sin x = \pm \frac{4}{5} $
Since the angle $x$ lies in the third quadrant, the value of $\sin x$ is negative in this quadrant.
Therefore, we choose the negative value:
$ \sin x = -\frac{4}{5} $
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What is sin 2α equal to?
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