Question

2arccos(1/2)+arctg(1)+arsin√2/2

Answer 1

2 arccos(1/2) + arctg(1) + arcsin(√2/2) = 2π/3 + π/4 + π/4 = 2π/3 + 2π/4 = 14π/12 = 7π/6

Answer 2

To simplify the expression, let's start by evaluating each individual term step by step:

1. 2arccos(1/2): The value of arccos(1/2) is the angle whose cosine is 1/2. The principal value of arccos(1/2) is π/3 radians or 60 degrees. Therefore, 2arccos(1/2) = 2 * π/3 = 2π/3.

2. arctg(1): The value of arctg(1) is the angle whose tangent is 1. The principal value of arctg(1) is π/4 radians or 45 degrees.

3. arsin(√2/2): The value of arcsin(√2/2) is the angle whose sine is √2/2. The principal value of arcsin(√2/2) is π/4 radians or 45 degrees.

Now, let's add these values together:

2π/3 + π/4 + π/4

To add fractions, we need a common denominator, which is 12 in this case:

(8π + 3π + 3π) / 12

Combine the numerators:

14π / 12

Now, simplify the fraction:

π / (12 / 14)

To reduce the fraction, find the greatest common divisor (GCD) of 12 and 14, which is 2:

π / (6 / 7)

Finally, multiply the numerator by the reciprocal of the denominator:

π * (7 / 6)

The simplified expression is:

(7π / 6)