Question

Sin(12pi/5)-3sin(3pi/5) решите пожалуйста

Answer 1

А) Решите уравнение 3sin2x + 3cos2x = 4
б) Найдите все корни этого уравнения, принадлежащие отрезку [2π; 7π/2]

вроде правельно 

Answer 2

Solving the Equation

To solve the equation sin(12pi/5) - 3sin(3pi/5), we can use the trigonometric identities and properties to simplify and find the solution.

Let's break down the equation step by step:

1. sin(12pi/5): We can rewrite this as sin(2pi - 3pi/5) using the periodicity of the sine function. This is because the sine function has a period of 2pi, so adding or subtracting multiples of 2pi does not change the value. Therefore, sin(12pi/5) = sin(2pi - 3pi/5).

2. 3sin(3pi/5): We can rewrite this as 3sin(pi - 3pi/5) using the periodicity of the sine function. Again, this is because the sine function has a period of 2pi, so adding or subtracting multiples of 2pi does not change the value. Therefore, 3sin(3pi/5) = 3sin(pi - 3pi/5).

Now, let's simplify the equation further:

sin(2pi - 3pi/5) - 3sin(pi - 3pi/5)

Using the angle subtraction formula for sine, we have:

sin(2pi)cos(3pi/5) - cos(2pi)sin(3pi/5) - 3sin(pi)cos(3pi/5) + 3cos(pi)sin(3pi/5)

Since sin(2pi) = 0 and cos(2pi) = 1, and sin(pi) = 0 and cos(pi) = -1, we can simplify the equation further:

0*cos(3pi/5) - 1*sin(3pi/5) - 3*0*cos(3pi/5) + 3*(-1)*sin(3pi/5)

Simplifying the equation gives us:

-sin(3pi/5) + 3sin(3pi/5)

Combining like terms, we get:

2sin(3pi/5)

Therefore, the simplified equation is 2sin(3pi/5).

Now, let's find the solution for 2sin(3pi/5).

Using the trigonometric identity sin(x) = 1 for x = pi/2 + 2pi*n, where n is an integer, we can solve for 2sin(3pi/5):

2sin(3pi/5) = 2 * 1 = 2

Therefore, the solution for the equation sin(12pi/5) - 3sin(3pi/5) is 2.

Conclusion

The solution for the equation sin(12pi/5) - 3sin(3pi/5) is 2.