Find Exact Value of Each Expression

**magentarita** · August 29th 2008, 02:43 AM

Find exact value of each expression.

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

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

**Moo** · August 29th 2008, 03:10 AM

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 : $\cos(\arcsin(x))=\sqrt{1-x^2}$

**magentarita** · August 29th 2008, 04:15 AM

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.

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