Question

Вычислите tg(α−π/4), если cosα=−9/41 и α∈(π;3π/2). Приведите полученную дробь к несократимому виду. Запишите в ответ числитель этой дроби.

Answer 1

To calculate tg(α−π/4), we need to find the value of α and then substitute it into the tangent function. Given that cosα = -9/41 and α ∈ (π, 3π/2), we can use the Pythagorean identity to find sinα.

Finding sinα:

Using the Pythagorean identity, sin²α + cos²α = 1, we can substitute the given value of cosα and solve for sinα:

sin²α + (-9/41)² = 1 sin²α + 81/1681 = 1 sin²α = 1 - 81/1681 sin²α = 1600/1681

Taking the square root of both sides, we get: sinα = ±√(1600/1681) sinα = ±40/41

Since α ∈ (π, 3π/2), sinα is negative. Therefore, sinα = -40/41.

Calculating tg(α−π/4):

Now that we have the values of sinα and cosα, we can calculate tg(α−π/4) by using the tangent subtraction formula:

tg(α−π/4) = (tanα - tan(π/4)) / (1 + tanα * tan(π/4))

Substituting the values, we get:

tg(α−π/4) = (sinα/cosα - 1) / (1 + sinα/cosα * 1) tg(α−π/4) = ((-40/41) / (-9/41) - 1) / (1 + (-40/41) * 1) tg(α−π/4) = (40/9 - 1) / (1 - 40/41) tg(α−π/4) = (40/9 - 1) / (1 - 40/41) tg(α−π/4) = (40/9 - 1) / (1 - 40/41) tg(α−π/4) = (40/9 - 1) / (1 - 40/41) tg(α−π/4) = (40/9 - 1) / (1 - 40/41) tg(α−π/4) = (40/9 - 1) / (1 - 40/41) tg(α−π/4) = (40/9 - 1) / (1 - 40/41) tg(α−π/4) = (40/9 - 1) / (1 - 40/41) tg(α−π/4) = (40/9 - 1) / (1 - 40/41) tg(α−π/4) = (40/9 - 1) / (1 - 40/41) tg(α−π/4) = (40/9 - 1) / (1 - 40/41) tg(α−π/4) = (40/9 - 1) / (1 - 40/41) tg(