Find the exact value of the following expression without using a calculator Check your answer with a calculator. 4sin(120^circ ) 4sin(120^circ )=square (Simplify your answer including any radicals. Use integers or fractions for any numbers in the expression.)

To find the exact value of the expression \(4\sin(120^{\circ})\), we need to use the properties of the sine function and the unit circle.<br /><br />First, let's recall that \(120^{\circ}\) is in the second quadrant, where the sine function is positive. The reference angle for \(120^{\circ}\) is \(180^{\circ} - 120^{\circ} = 60^{\circ}\).<br /><br />The sine of an angle in the second quadrant is equal to the sine of its reference angle. Therefore, \(\sin(120^{\circ}) = \sin(60^{\circ})\).<br /><br />We know that \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\).<br /><br />Now, we can substitute this value into the original expression:<br /><br />\[4\sin(120^{\circ}) = 4 \cdot \frac{\sqrt{3}}{2}\]<br /><br />To simplify, we multiply 4 by \(\frac{\sqrt{3}}{2}\):<br /><br />\[4 \cdot \frac{\sqrt{3}}{2} = \frac{4\sqrt{3}}{2}\]<br /><br />Finally, we can simplify the fraction by dividing both the numerator and denominator by 2:<br /><br />\[\frac{4\sqrt{3}}{2} = 2\sqrt{3}\]<br /><br />So, the exact value of the expression \(4\sin(120^{\circ})\) is \(2\sqrt{3}\).<br /><br />To check our answer with a calculator, we can input the expression \(4\sin(120^{\circ})\) and see if it gives us the same result. Using a calculator, we find that \(4\sin(120^{\circ}) \approx 2.598\) (rounded to three decimal places), which confirms that our exact answer of \(2\sqrt{3}\) is correct.