Question

4 sin 3x+ 3 cos 3x =5,2​

Answer 1

Ответ: 4sin3x - 3cos3x=5*sin(3x-arctg(3/4))=5/2 => sin(3x-arctg(3/4))=1/2

3x=arctg(3/4) +((-1)^k)*(pi/6) +(pi)*k

Остается поделить на три.

Пошаговое объяснение:

Answer 2

To solve the equation 4sin(3x) + 3cos(3x) = 5/2, you can use trigonometric identities to simplify it. One common approach is to rewrite sin(3x) and cos(3x) in terms of a single trigonometric function, typically by using the identity sin^2(x) + cos^2(x) = 1.

Let's start by squaring both sides of the equation:

(4sin(3x) + 3cos(3x))^2 = (5/2)^2

Expand the left side of the equation using the trigonometric identity sin^2(x) + cos^2(x) = 1 and the double angle formulas for sin(2θ) and cos(2θ):

(4sin(3x))^2 + 2(4sin(3x))(3cos(3x)) + (3cos(3x))^2 = (5/2)^2

Now, apply the double angle formulas:

(4sin(3x))^2 + 2(4sin(3x))(3cos(3x)) + (3cos(3x))^2 = 25/4

Expand further:

16sin^2(3x) + 24sin(3x)cos(3x) + 9cos^2(3x) = 25/4

Now, use the identity sin^2(θ) + cos^2(θ) = 1 to replace sin^2(3x) + cos^2(3x) with 1:

16(1 - cos^2(3x)) + 24sin(3x)cos(3x) + 9cos^2(3x) = 25/4

Now, you have a quadratic equation in terms of cos(3x). Rearrange it and multiply through by 4 to get rid of the fraction:

64 - 64cos^2(3x) + 96sin(3x)cos(3x) + 36cos^2(3x) = 25

Combine like terms:

-28cos^2(3x) + 96sin(3x)cos(3x) + 39 = 25

Rearrange the equation:

-28cos^2(3x) + 96sin(3x)cos(3x) + 25 = 0

Now, you have a quadratic equation in terms of cos(3x). You can solve this equation for cos(3x) using the quadratic formula:

cos(3x) = (-B ± √(B² - 4AC)) / 2A

In this case, A = -28, B = 96sin(3x), and C = 25. Plug these values into the quadratic formula and solve for cos(3x).

cos(3x) = (-96sin(3x) ± √((96sin(3x))² - 4(-28)(25))) / (2(-28))

cos(3x) = (-96sin(3x) ± √(9216sin^2(3x) + 2800)) / (-56)

Now, you have two possible solutions for cos(3x), which will lead to two different solutions for x:

1. cos(3x) = (-96sin(3x) + √(9216sin^2(3x) + 2800)) / (-56)
2. cos(3x) = (-96sin(3x) - √(9216sin^2(3x) + 2800)) / (-56)

To find specific values of x that satisfy the original equation, you'll need to use numerical methods or a calculator because solving for sin(3x) and cos(3x) directly can be quite complex.