Find Equivalent Angle: To find an equivalent angle for (5π)/(4) that lies within the range of [−π/2,π/2], we can subtract multiples of 2π until the angle is within the desired range. However, since (5π)/(4) is only π/4 beyond π, we can also find a coterminal angle by subtracting π from (5π)/(4), which gives us (5π)/(4)−π=(5π)/(4)−(4π)/(4)=(π)/(4). But we need to consider that the sine function is negative in the third quadrant where (5π)/(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]0.
Evaluate arcsin(sin): Now we can evaluate arcsin(sin(45π)) by using the equivalent angle we found in the previous step. Since sin(45π) is equal to sin(−4π), and −4π is within the range of arcsin, we have arcsin(sin(45π))=arcsin(sin(−4π))=−4π.
More problems from Find derivatives of inverse trigonometric functions