Simplify cos[arccsc((2/3)/3)].
a)no solution
b)sqrt3/2
c)1/2

1/2

Step-by-step explanation:

We have .

Let .

This implies .

Use that sine and cosecant are reciprocals.

Now I'm going to rationalize the denominator there by multiply numerator and denominator by :

Reduce the fraction:

Now I'm going to use a Pythagorean Identity: .

This will give me the value of cos(u) which would give me the answer to my question if it exists.

Replace with in:

Subtract 3/4 on both sides:

Square root both sides:

(since 1/2*1/2=1/4 or -1/2*-1/2=1/4)

Now we must decide between the positive or the negative.

It depends where u lies.  What quadrant? Hopefully it lays between 0 and .  Otherwise, it doesn't exist (unless you have a different definition for arc function).

So u led to this equation earlier:

arcsin( ) only has outputs between and .

This would have to be in the first quadrant because we have only positive sine values there.

So this means cos(u)=1/2 and not -1/2 because we are using that u is in the 1st quadrant.

Remember u was .

So we have actually evaluated

without a calculator.

The value is 1/2.

you have to put the graph on here with the question

step-by-step explanation: