Question

4 sin α - 3 cos α/ 2 sin α + 5 cos α, если tg α = -3

Answer 1

Ответ:

решение смотри на фотографии

Объяснение:

Answer 2

To simplify the expression 4sin(α) - (3cos(α))/(2sin(α) + 5cos(α)), we can use trigonometric identities. Since we are given that tan(α) = -3, we can use this information to find the values of sin(α) and cos(α).

Recall that: tan(α) = sin(α)/cos(α)

Given that tan(α) = -3, we can write: -3 = sin(α)/cos(α)

Now, let's find sin(α) and cos(α): sin(α) = -3cos(α)

Now, we can substitute the value of sin(α) into the expression to simplify it:

4sin(α) - (3cos(α))/(2sin(α) + 5cos(α))

= 4(-3cos(α)) - (3cos(α))/(2(-3cos(α)) + 5cos(α))

= -12cos(α) - (3cos(α))/(-6cos(α) + 5cos(α))

= -12cos(α) - (3cos(α))/(-6cos(α) + 5cos(α))

= -12cos(α) - (3cos(α))/(-1)

= -12cos(α) + 3cos(α)

= -9cos(α)

So, the simplified expression is -9cos(α).

Now, to find the value of cos(α), we can use the fact that sin(α)^2 + cos(α)^2 = 1: (-3cos(α))^2 + cos(α)^2 = 1

9cos(α)^2 + cos(α)^2 = 1

10cos(α)^2 = 1

cos(α)^2 = 1/10

cos(α) = ±√(1/10)

Since tan(α) is negative, we are dealing with the second or third quadrant, where cosine is negative. So, cos(α) = -√(1/10).

Now, we can calculate the value of the simplified expression:

-9cos(α) = -9 * (-√(1/10)) = 9√10 / 10

Therefore, the value of the expression is 9√10 / 10 (approximately 2.83).