Question

Найдите значение sin(Ф-60°), если sinФ= -3/4 и 180°<Ф<270°

Answer 1

Смотрите прикреплённую картинку

Answer 2

To find the value of sin(Φ - 60°) given that sinΦ = -3/4 and 180° < Φ < 270°, we can use the trigonometric identity for the sine of the difference of two angles:

sin(Φ - 60°) = sinΦ * cos(60°) - cosΦ * sin(60°)

We already know that sinΦ = -3/4, but we need to find cosΦ.

Since Φ is in the third quadrant (180° < Φ < 270°), the reference angle, which is the angle between the terminal side of Φ and the x-axis, is 180° - Φ.

Now, we know sinΦ = -3/4 and cosΦ is positive in the third quadrant, so cosΦ = √(1 - sin^2Φ). Let's calculate it:

cosΦ = √(1 - (-3/4)^2) cosΦ = √(1 - 9/16) cosΦ = √(16/16 - 9/16) cosΦ = √(7/16) cosΦ = √7/4

Now, we need to find sin(60°) and cos(60°):

sin(60°) = √3/2 cos(60°) = 1/2

Now we can calculate sin(Φ - 60°):

sin(Φ - 60°) = sinΦ * cos(60°) - cosΦ * sin(60°) sin(Φ - 60°) = (-3/4) * (1/2) - (√7/4) * (√3/2) sin(Φ - 60°) = (-3/8) - (√21/8) sin(Φ - 60°) = (-3 - √21) / 8

So the value of sin(Φ - 60°) is (-3 - √21) / 8.