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.First, let's recall that is in the second quadrant, where the sine function is positive. The reference angle for is .The sine of an angle in the second quadrant is equal to the sine of its reference angle. Therefore, \(\sin(120^{\circ}) = \sin(60^{\circ})\).We know that \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\).Now, we can substitute this value into the original expression: To simplify, we multiply 4 by : Finally, we can simplify the fraction by dividing both the numerator and denominator by 2: So, the exact value of the expression \(4\sin(120^{\circ})\) is .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 is correct.