Question

Дано cos альфа -5\13 где п\2 <а<п найдите ctg 2a

Answer 1

Так как вторая четверть, то sin(a) положительный. т.е. sin(a)=12/13, и тогда ctg=(-5/13)/(12/13)=-5/12

Answer 2

Solving for ctg(2a) given cos(α) = -5/13 where π/2 < α < π

To find ctg(2a), we can use the relationship between ctg(2a) and cos(2a). First, we need to find the value of cos(2a) using the given information about cos(α).

Finding the value of cos(2a)

We know the double angle identity for cosine is:

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

Given that cos(α) = -5/13, we can use this information to find cos(a) and then substitute it into the double angle identity to find cos(2a).

Using the Pythagorean identity: sin^2(a) + cos^2(a) = 1, we can find sin(a) and then cos(a).

From the given information, we have: cos(α) = -5/13

Using the Pythagorean identity, we can find sin(α):

sin^2(α) = 1 - cos^2(α)

sin^2(α) = 1 - (-5/13)^2

sin^2(α) = 1 - 25/169

sin^2(α) = 144/169

sin(α) = ±12/13

Now that we have sin(α) and cos(α), we can find cos(2a):

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

cos(2a) = 2 * (-5/13)^2 - 1

cos(2a) = 2 * (25/169) - 1

cos(2a) = 50/169 - 1

cos(2a) = -119/169

Finding ctg(2a)

Now that we have the value of cos(2a), we can find ctg(2a) using the relationship between ctg(2a) and cos(2a).

The relationship between ctg(2a) and cos(2a) is:

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

Substituting the value of cos(2a) and sin(2a) we found earlier:

ctg(2a) = (-119/169) / (±12/13)

ctg(2a) = (-119/169) * (13/±12)

ctg(2a) = -119/(-169 * 12) or ctg(2a) = -119/(169 * 12)

Therefore, ctg(2a) = -119/(169 * 12).