To simplify this expression, we first need to analyze the values of the trigonometric functions at the given angles.
arccos(√3/2):This corresponds to an angle in the first quadrant where the cosine value is √3/2. This angle is π/6 radians or 30 degrees.
arcsin(-√2/2):This corresponds to an angle in the fourth quadrant where the sine value is -√2/2. This angle is -3π/4 radians or -135 degrees.
Now we can substitute these angles back into the expression:
4arccos(√3/2) - 1/2arcsin(-√2/2)= 4(π/6) - 1/2(-3π/4)= 4π/6 + 3π/8= (32π + 9π) / 48= 41π/48
Therefore, the simplified expression is 41π/48.
To simplify this expression, we first need to analyze the values of the trigonometric functions at the given angles.
arccos(√3/2):
This corresponds to an angle in the first quadrant where the cosine value is √3/2. This angle is π/6 radians or 30 degrees.
arcsin(-√2/2):
This corresponds to an angle in the fourth quadrant where the sine value is -√2/2. This angle is -3π/4 radians or -135 degrees.
Now we can substitute these angles back into the expression:
4arccos(√3/2) - 1/2arcsin(-√2/2)
= 4(π/6) - 1/2(-3π/4)
= 4π/6 + 3π/8
= (32π + 9π) / 48
= 41π/48
Therefore, the simplified expression is 41π/48.