Question

Sin(arcctg (1/2) - arcctg(-1/3))^2

Answer 1

Ответ:

Пошаговое объяснение:

Answer 2

To simplify the expression Sin(arcctg(1/2) - arcctg(-1/3))^2, we can use the properties of trigonometric functions and the inverse trigonometric functions.

Let's start by simplifying the expression inside the sine function: arcctg(1/2) - arcctg(-1/3).

The inverse cotangent function (arcctg or arccot) is the inverse of the cotangent function (ctg or cot). We can use the following trigonometric identity:

arcctg(x) = arctan(1/x)

Therefore, we have:

arcctg(1/2) = arctan(1/(1/2)) = arctan(2)

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

Now, let's substitute these values back into the original expression:

Sin(arcctg(1/2) - arcctg(-1/3))^2 = Sin(arctan(2) - arctan(-3))^2

Using the subtraction formula for tangent:

tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a) * tan(b))

We can rewrite the expression as:

Sin(arctan(2) - arctan(-3))^2 = Sin(arctan(2 + 3 / (1 - 2 * -3)))^2 = Sin(arctan(2 + 3/7))^2 = Sin(arctan(17/7))^2

Now, we can simplify further:

Using the identity:

Sin(arctan(x)) = x / sqrt(1 + x^2)

We have:

Sin(arctan(17/7))^2 = (17/7) / sqrt(1 + (17/7)^2))^2

Simplifying the expression inside the square root:

sqrt(1 + (17/7)^2) = sqrt(1 + 289/49) = sqrt(338/49) = sqrt(338) / 7

Substituting this back into the original expression:

(17/7) / sqrt(1 + (17/7)^2))^2 = (17/7) / (sqrt(338) / 7))^2 = (17/7) / (sqrt(338) / 7)^2 = (17/7) / (338/49) = (17/7) * (49/338) = 833/338

Therefore, Sin(arcctg(1/2) - arcctg(-1/3))^2 simplifies to 833/338.