Final Answer:
(a) \( \arcsin \left(\sin \left(\frac{5 \pi}{3}\right)\right) = -\frac{\pi}{3} \).
(b) \( \sin \left(\frac{1}{2} \arcsin \left(\frac{1}{2}\right)\right) = \frac{\sqrt{2}}{2} \).
Step-by-step explanation:
(a)
To find \( \arcsin \left(\sin \left(\frac{5 \pi}{3}\right)\right) \), we first evaluate the inner function: \( \sin \left(\frac{5 \pi}{3}\right) \). The angle \( \frac{5 \pi}{3} \) is in the third quadrant, where the sine function is negative. Since \( \sin \left(\frac{5 \pi}{3}\right) = \sin \left(\pi + \frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) \), we can simplify the expression to \( -\frac{\sqrt{3}}{2} \). The arcsin of this value is \( -\frac{\pi}{3} \).
(b)
For \( \sin \left(\frac{1}{2} \arcsin \left(\frac{1}{2}\right)\right) \), we use the property \( \sin\left(\arcsin(x)\right) = x \) to simplify the expression. Thus, \( \sin \left(\frac{1}{2} \arcsin \left(\frac{1}{2}\right)\right) = \sin \left(\frac{\pi}{6}\right) \), which is equal to \( \frac{1}{2} \).
Step-by-step explanation:
In part (a), the angle \( \frac{5 \pi}{3} \) is represented in radians and is converted to its equivalent angle in the unit circle. The sine of this angle is determined using the properties of the sine function in different quadrants. The negative sign indicates the third quadrant. The arcsin function then gives the angle whose sine is \( -\frac{\sqrt{3}}{2} \), which is \( -\frac{\pi}{3} \).
In part (b), the expression involves the composition of arcsin and sin functions. Using the property \( \sin\left(\arcsin(x)\right) = x \), the expression is simplified to \( \sin \left(\frac{\pi}{6}\right) \), which is known to be \( \frac{1}{2} \).