Question

Найдите значение sin(φ-30º) если cosφ= корень 13/4 и 270º<φ<360º

Answer 1

Вроде так, если значение косинуса изначально я верно поняла.

Answer 2

To find the value of sin(φ - 30º), we first need to determine the value of sin(φ) using the given information and then apply the trigonometric identity for sin(φ - 30º).

Given: cosφ = √(13/4) 270º < φ < 360º

We know that cos(φ) = adjacent / hypotenuse in a right-angled triangle. So, let's assume a right-angled triangle with an angle φ, where cos(φ) = √(13/4).

The adjacent side of the triangle is √13, and the hypotenuse is 4. To find the opposite side, we can use the Pythagorean theorem:

opposite^2 + adjacent^2 = hypotenuse^2 opposite^2 + (√13)^2 = 4^2 opposite^2 + 13 = 16 opposite^2 = 16 - 13 opposite^2 = 3 opposite = √3

Now, we have the values of the opposite and adjacent sides of the triangle:

opposite = √3 adjacent = √13 hypotenuse = 4

Next, we'll use the trigonometric identity for sin(φ - 30º):

sin(φ - 30º) = sinφ * cos30º - cosφ * sin30º

Since we have the values of sinφ and cosφ, we can substitute them:

sin(φ - 30º) = (√3/4) * (√3/2) - (√13/4) * (1/2)

Now, let's simplify:

sin(φ - 30º) = (3/4) * (√3/2) - (√13/4) * (1/2) sin(φ - 30º) = (3√3/8) - (√13/8) sin(φ - 30º) = (3√3 - √13) / 8

So, the value of sin(φ - 30º) when 270º < φ < 360º is (3√3 - √13) / 8.