Hello KAY03 Originally Posted by

**KAY03** I am having trouble with this problem...It goes as follows:

Find the exact value of:

sin(2x), cos(2x), and tan(2x) if cos(x) = -4/5, [pi]/2 <x<[pi].

It gives me a hint saying( sin2x=2sinxcosx and cos(2x)=2cos^2 (X-1)

I have no idea how to do this

Anyone care to help?

First, note that, if $\displaystyle \pi/2 <x<\pi/2$, then $\displaystyle \sin x$ is positive.

Then, using $\displaystyle \sin^2x =1-\cos^2x$, plug in the value $\displaystyle \cos x = -\tfrac45$:$\displaystyle \sin^2x=1 - \tfrac{16}{25} $$\displaystyle =\tfrac{9}{25}$

$\displaystyle \Rightarrow\sin x = \tfrac35$ (taking the positive square root, since, as we said, $\displaystyle \sin x > 0$)

Now you can simply plug in the values $\displaystyle \sin x = \tfrac35$ and $\displaystyle \cos x = -\tfrac45$ into the formulae that you've been given. The first one is:

$\displaystyle \sin 2x =2\sin x \cos x$$\displaystyle =2\cdot\tfrac35\cdot(-\tfrac45)$

$\displaystyle = ...$ ?

Do $\displaystyle \cos 2x$ in the same way, using the formula:

$\displaystyle \cos2x=2\cos^2x -1$

Finally, use $\displaystyle \tan2x = \frac{\sin2x}{\cos2x}$ to find $\displaystyle \tan2x$.

Can you complete them now?

Grandad