ind the exact value of the following function without a calculator sin((pi )/(4)) sin((pi )/(4))=square (Simplify your answer.including any radicals. Use integers or fractions for any numbers in the expression.)

To find the exact value of $\sin(\frac{\pi}{4})$, we can use the unit circle and the properties of a 45-45-90 triangle.<br /><br />Step 1: Recall that the angle $\frac{\pi}{4}$ radians is equivalent to 45 degrees.<br /><br />Step 2: In a 45-45-90 triangle, the lengths of the sides are in the ratio $1:1:\sqrt{2}$. <br /><br />Step 3: The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.<br /><br />Step 4: In this case, the side opposite the angle $\frac{\pi}{4}$ is 1 and the hypotenuse is $\sqrt{2}$.<br /><br />Step 5: Therefore, $\sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$.<br /><br />Step 6: To simplify the expression, we can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$.<br /><br />Step 7: $\sin(\frac{\4}) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.<br /><br />Answer: $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$.