Given #csctheta^circ=sqrt3/2# and #sectheta^circ=sqrt3/3#, how do you find #sintheta#?

Given ( \csc(\theta) = \frac{\sqrt{3}}{2} ) and ( \sec(\theta) = \frac{\sqrt{3}}{3} ), you can find ( \sin(\theta) ) using the following steps:

1. Recall that ( \csc(\theta) ) is the reciprocal of ( \sin(\theta) ), and ( \sec(\theta) ) is the reciprocal of ( \cos(\theta) ).
2. Use the given values to find ( \sin(\theta) ) and ( \cos(\theta) ) reciprocals.
3. Apply the Pythagorean identity ( \sin^2(\theta) + \cos^2(\theta) = 1 ) to find ( \sin(\theta) ).
4. Once you have ( \sin(\theta) ), verify its sign using the given values to ensure it is correct.

Calculating ( \sin(\theta) ) using the reciprocals of ( \csc(\theta) ) and ( \sec(\theta) ), you would have:

[ \sin(\theta) = \frac{1}{\csc(\theta)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} ]

Then, rationalize the denominator:

[ \sin(\theta) = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} ]

Thus, ( \sin(\theta) = \frac{2\sqrt{3}}{3} ).