# Thread: Find Exact Value of Each Expression

1. ## Find Exact Value of Each Expression

Find exact value of each expression.

(1) sin[2arcsin(sqrt{3}/2)]

(2) cos^2[1/2arcsin(3/5)]

2. Originally Posted by magentarita
Find exact value of each expression.

(1) sin[2arcsin(sqrt{3}/2)]
$\sin(2x)=2 \sin(x) \cos(x)$

Here, $x=\arcsin \left(\frac{\sqrt{3}}{2}\right)$

Remember the formula $\cos(\arcsin(x))=\sqrt{1-x^2}$

----> $\sin \left(2 \arcsin \left(\frac{\sqrt{3}}{2}\right) \right)=2 \frac{\sqrt{3}}{2} \sqrt{1-\left(\frac{\sqrt{3}}{2}\right)^2}$

etc...

(2) cos^2[1/2arcsin(3/5)]
We know that $\cos(2x)=2 \cos^2(x)-1 \implies \cos(x)=2 \cos^2 \tfrac x2 -1 \implies \cos^2 \tfrac x2=\tfrac{\cos(x)+1}{2}$

Now, you just have to calculate $\cos(\arcsin(\tfrac 35))$
Once again, this formula : $\cos(\arcsin(x))=\sqrt{1-x^2}$

3. ## Fabulous...

Originally Posted by Moo
$\sin(2x)=2 \sin(x) \cos(x)$

Here, $x=\arcsin \left(\frac{\sqrt{3}}{2}\right)$

Remember the formula $\cos(\arcsin(x))=\sqrt{1-x^2}$

----> $\sin \left(2 \arcsin \left(\frac{\sqrt{3}}{2}\right) \right)=2 \frac{\sqrt{3}}{2} \sqrt{1-\left(\frac{\sqrt{3}}{2}\right)^2}$

etc...

We know that $\cos(2x)=2 \cos^2(x)-1 \implies \cos(x)=2 \cos^2 \tfrac x2 -1 \implies \cos^2 \tfrac x2=\tfrac{\cos(x)+1}{2}$

Now, you just have to calculate $\cos(\arcsin(\tfrac 35))$
Once again, this formula : $\cos(\arcsin(x))=\sqrt{1-x^2}$
Fabulous work as always.