Question

Ctg a/2 = 2 найдите ctg a

Answer 1

1) Ctg a = 1/tg a
2) tg 2a = 2 tg a/( 1 - tg²a)
Ctg a/2 = 2 ⇒ tg a/2 = 1/2
tg a = 2 tg a/2/(1 - tg ² a/2) = 2·1/2 /(1 - 1/4)= 1/ 3/4= 4/3
Сtg a = 3/4

Answer 2

To find the value of ctg(a), we can use the given equation:

ctg(a/2) = 2

We know that cotangent (ctg) is the reciprocal of the tangent function, so we can rewrite the equation as:

1/tan(a/2) = 2

To solve for ctg(a), we need to find the value of tan(a/2) first. We can do this by taking the reciprocal of both sides:

tan(a/2) = 1/2

Now, we can use the half-angle identity for the tangent function, which states:

tan(a/2) = sin(a) / (1 + cos(a))

Plugging in the value we have:

1/2 = sin(a) / (1 + cos(a))

To simplify further, we can cross-multiply:

1 + cos(a) = 2sin(a)

Next, let's use the Pythagorean identity for sine and cosine:

sin^2(a) + cos^2(a) = 1

Rearranging this equation, we get:

1 - cos^2(a) = sin^2(a)

Now, we can substitute this identity into the previous equation:

1 + cos(a) = 2(1 - cos^2(a))

Expanding and simplifying:

1 + cos(a) = 2 - 2cos^2(a)

Rearranging again:

2cos^2(a) + cos(a) - 1 = 0

Now, we have a quadratic equation in terms of cos(a). Let's solve for cos(a) using the quadratic formula:

cos(a) = (-1 ± √(1^2 - 4(2)(-1))) / (2(2))

cos(a) = (-1 ± √(1 + 8)) / 4

cos(a) = (-1 ± √9) / 4

cos(a) = (-1 ± 3) / 4

Now, we have two possible values for cos(a):

cos(a) = (-1 + 3) / 4 = 2 / 4 = 1/2

cos(a) = (-1 - 3) / 4 = -4 / 4 = -1

Since the cosine function is positive in the first and fourth quadrants, we can conclude that cos(a) = 1/2.

Now, we can substitute this value back into the equation:

1 + cos(a) = 2sin(a)

1 + 1/2 = 2sin(a)

3/2 = 2sin(a)

sin(a) = (3/2) / 2 = 3/4

Finally, we can find the value of ctg(a) by taking the reciprocal of the tangent function:

ctg(a) = 1/tan(a) = 1/(sin(a)/cos(a)) = cos(a)/sin(a) = (1/2) / (3/4) = (1/2) * (4/3) = 2/3

Therefore, ctg(a) is equal to 2/3.