Question

Find the exact value pf cos 2x if sin x = - 4/5 (Quadrant III)

Answer 1

Answer:

$\displaystyle \cos(2x)=-\frac{7}{25}$

Step-by-step explanation:

We want to find:

$\displaystyle \cos(2x)\text{ given that }\sin(x)=-\frac{4}{5}$

And x is in QIII.

Recall that sine is the ratio of the opposite side to the hypotenuse.

Therefore, the adjacent side is (we can ignore the negative for now).

$a=\sqrt{5^2-4^2}=\sqrt{25-16}=\sqrt{9}=3$

So, the adjacent side is 3, the opposite side is 4, and the hypotenuse is 5.

Using the double angle identity, we can rewrite:

$\cos(2x)=\cos^2(x)-\sin^2(x)$

Since x is in QIII, cosine is positive, sine is negative, and tangent is negative.

Using the above values, we can conclude that:

$\displaystyle \cos(x)=\frac{3}{5}\text{ and } \sin(x)=-\frac{4}{5}$

Substitute:

$\displaystyle \cos(2x)=\Big(\frac{3}{5}\Big)^2-\Big(-\frac{4}{5}\Big)^2$

Evaluate:

$\displaystyle \cos(2x)=\frac{9}{25}-\frac{16}{25}=-\frac{7}{25}$