Question

Sin²(arcctg(1/2) - arctg(-1/3))

Answer 1

Ответ:

0,02

Пошаговое объяснение:

sin^2(arctg(1/7)=((1/7)/(sqrt(1+1/49)))^2=1/49:50/49=1/50=0,02

Answer 2

To simplify the expression sin²(arctan(1/2) - arctan(-1/3)), we can use the trigonometric identity for the difference of arctan angles:

arctan(a) - arctan(b) = arctan((a - b) / (1 + ab))

In this case, a = 1/2 and b = -1/3, so we have:

arctan(1/2) - arctan(-1/3) = arctan((1/2 - (-1/3)) / (1 + (1/2)(-1/3)))

Now, let's calculate the values in the arctan expression:

1/2 - (-1/3) = 1/2 + 1/3 = 5/6 1 + (1/2)(-1/3) = 1 - 1/6 = 5/6

So, we have:

arctan(1/2) - arctan(-1/3) = arctan((5/6) / (5/6))

Now, the fractions cancel out:

arctan(1/2) - arctan(-1/3) = arctan(1)

Now, we need to find sin²(arctan(1)). To do that, we can use the fact that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side in a right triangle. So, if we consider a right triangle with an angle of arctan(1), the opposite side and the adjacent side will have the same length.

Let's assume that the opposite and adjacent sides both have a length of 1 unit. Then, by the Pythagorean theorem, the hypotenuse will have a length of √2 units.

Now, we can calculate sin(arctan(1)):

sin(arctan(1)) = opposite side / hypotenuse = 1/√2

Finally, square sin(arctan(1)) to get sin²(arctan(1)):

sin²(arctan(1)) = (1/√2)² = 1/2

So, sin²(arctan(1/2) - arctan(-1/3)) = 1/2.