To find the value of arccos(-1/2) - arcsin(√3/2), we need to first determine the angles that have cosine equal to -1/2 and sine equal to √3/2.
arccos(-1/2): The cosine of an angle is equal to -1/2 in the second and third quadrants. In the second quadrant, the reference angle is π/3, so the angle in that quadrant will be 2π/3. In the third quadrant, the reference angle is also π/3, so the angle in that quadrant will be 4π/3.
arcsin(√3/2): The sine of an angle is equal to √3/2 in the first and second quadrants. In the first quadrant, the reference angle is π/3, so the angle in that quadrant will be π/3. In the second quadrant, the reference angle is also π/3, so the angle in that quadrant will be 2π/3.
Now we can calculate the value of arccos(-1/2) - arcsin(√3/2):
To find the value of arccos(-1/2) - arcsin(√3/2), we need to first determine the angles that have cosine equal to -1/2 and sine equal to √3/2.
arccos(-1/2): The cosine of an angle is equal to -1/2 in the second and third quadrants. In the second quadrant, the reference angle is π/3, so the angle in that quadrant will be 2π/3. In the third quadrant, the reference angle is also π/3, so the angle in that quadrant will be 4π/3.
arcsin(√3/2): The sine of an angle is equal to √3/2 in the first and second quadrants. In the first quadrant, the reference angle is π/3, so the angle in that quadrant will be π/3. In the second quadrant, the reference angle is also π/3, so the angle in that quadrant will be 2π/3.
Now we can calculate the value of arccos(-1/2) - arcsin(√3/2):
arccos(-1/2) = 2π/3 or 4π/3 (because cos(2π/3) = cos(4π/3) = -1/2)
arcsin(√3/2) = π/3 or 2π/3 (because sin(π/3) = sin(2π/3) = √3/2)
Therefore, the possible values for arccos(-1/2) - arcsin(√3/2) are:
2π/3 - π/3 = π/32π/3 - 2π/3 = 04π/3 - π/3 = π4π/3 - 2π/3 = 2π/3So, arccos(-1/2) - arcsin(√3/2) can be equal to π/3, 0, π, or 2π/3.