arctg(-1) is the angle whose tangent is -1. Since tangent is negative in the second and fourth quadrants, and the tangent of -π/4 (or -45 degrees) is -1, we have:
arctg(-1) = -π/4
Next, let's find the value of arccos (√3/2):
arccos (√3/2) is the angle whose cosine is √3/2. Since cosine is positive in the first quadrant, and the cosine of π/6 (or 30 degrees) is √3/2, we have:
arccos(√3/2) = π/6
Therefore, arctg(-1) + arccos(√3/2) = -π/4 + π/6
To add these two angles, we need a common denominator, which is 12:
First, let's find the value of arctg(-1):
arctg(-1) is the angle whose tangent is -1. Since tangent is negative in the second and fourth quadrants, and the tangent of -π/4 (or -45 degrees) is -1, we have:
arctg(-1) = -π/4
Next, let's find the value of arccos (√3/2):
arccos (√3/2) is the angle whose cosine is √3/2. Since cosine is positive in the first quadrant, and the cosine of π/6 (or 30 degrees) is √3/2, we have:
arccos(√3/2) = π/6
Therefore, arctg(-1) + arccos(√3/2) = -π/4 + π/6
To add these two angles, we need a common denominator, which is 12:
arctg(-1) + arccos(√3/2) = -3π/12 + 2π/12 = -π/12
So, arctg(-1) + arccos(√3/2) = -π/12.