Question

Arc cos (-√3/2)-3/2arccos1/2+4arctg(-√3

Answer 1

Ответ:

arcsin -1 - 3\2 arccos 1\2 + 3 arctg (-1 корень из 3)=3π/2-3/2*π/6-3*π/3=

3π/2-3π/12-π=3π/12=π/4

Answer 2

To evaluate the expression, let's break it down step by step.

1. Arc cos (-√3/2): The range of the arccosine function is typically [0, π]. However, since the input is negative, we need to take into account the negative values in the range. The reference angle for -√3/2 is π/6. Thus, we have: Arc cos (-√3/2) = π - π/6 = 5π/6.

2. 3/2arccos(1/2): The arccosine of 1/2 is π/3. Multiply it by 3/2 to get: 3/2arccos(1/2) = 3/2 * π/3 = π/2.

3. 4arctg(-√3): The arctangent of -√3 is -π/3. Multiply it by 4 to get: 4arctg(-√3) = 4 * (-π/3) = -4π/3.

Now, we can substitute the values we found into the original expression:

Arc cos (-√3/2) - 3/2arccos(1/2) + 4arctg(-√3) = 5π/6 - π/2 - 4π/3.

To simplify this expression, we need to find a common denominator:

5π/6 = 15π/18 π/2 = 9π/18 4π/3 = 24π/18

Now, we can combine the fractions:

15π/18 - 9π/18 - 24π/18 = (15π - 9π - 24π) / 18 = (-18π) / 18 = -π.

Therefore, the value of the expression Arc cos (-√3/2) - 3/2arccos(1/2) + 4arctg(-√3) is -π.