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arcsin(sin((5pi)/(4)))

arcsin(sin(5π4)) \arcsin \left(\sin \left(\frac{5 \pi}{4}\right)\right)

Full solution

Q. arcsin(sin(5π4)) \arcsin \left(\sin \left(\frac{5 \pi}{4}\right)\right)
  1. Find Equivalent Angle: To find an equivalent angle for (5π)/(4)(5\pi)/(4) that lies within the range of [π/2,π/2][-\pi/2, \pi/2], we can subtract multiples of 2π2\pi until the angle is within the desired range. However, since (5π)/(4)(5\pi)/(4) is only π/4\pi/4 beyond π\pi, we can also find a coterminal angle by subtracting π\pi from (5π)/(4)(5\pi)/(4), which gives us (5π)/(4)π=(5π)/(4)(4π)/(4)=(π)/(4)(5\pi)/(4) - \pi = (5\pi)/(4) - (4\pi)/(4) = (\pi)/(4). But we need to consider that the sine function is negative in the third quadrant where (5π)/(4)(5\pi)/(4) lies, so we need to take the negative of the equivalent angle in the first quadrant to get the correct sign. Therefore, the equivalent angle is [π/2,π/2][-\pi/2, \pi/2]00.
  2. Evaluate arcsin(sin)\arcsin(\sin): Now we can evaluate arcsin(sin(5π4))\arcsin(\sin(\frac{5\pi}{4})) by using the equivalent angle we found in the previous step. Since sin(5π4)\sin(\frac{5\pi}{4}) is equal to sin(π4)\sin(-\frac{\pi}{4}), and π4-\frac{\pi}{4} is within the range of arcsin\arcsin, we have arcsin(sin(5π4))=arcsin(sin(π4))=π4\arcsin(\sin(\frac{5\pi}{4})) = \arcsin(\sin(-\frac{\pi}{4})) = -\frac{\pi}{4}.

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