Question

Sin x+ sin4x+sin7x=0

Answer 1

Ответ:

$sin x+sin 4x+sin 7x=0$

$sin4x+2sin ((x+7x)/2)cos ((x+7x) /2)=0$

$sin4x+2sin 4x cos4x=0$

$2sin4xcos4x= -sin4x$

$cos4x=-1/2$

$4x=4\pi /3$

$x=\pi/3$

Answer 2

The equation you've provided is a trigonometric equation involving the sum of three sine terms. The equation is:

sin(x) + sin(4x) + sin(7x) = 0

Solving trigonometric equations can sometimes be challenging, especially when dealing with multiple trigonometric functions. To solve this equation, you'll need to use trigonometric identities and techniques. One approach is to use the sum-to-product trigonometric identities.

One such identity is:

sin(A) + sin(B) = 2 * sin((A + B) / 2) * cos((A - B) / 2)

Applying this identity to the given equation:

sin(x) + sin(4x) = 2 * sin((x + 4x) / 2) * cos((x - 4x) / 2) = 2 * sin(2.5x) * cos(-1.5x)

sin(7x) can also be rewritten using this identity:

sin(7x) = sin(4x + 3x) = 2 * sin((4x + 3x) / 2) * cos((4x - 3x) / 2) = 2 * sin(3.5x) * cos(0.5x)

Now the equation becomes:

2 * sin(2.5x) * cos(-1.5x) + 2 * sin(3.5x) * cos(0.5x) = 0

Dividing both sides by 2:

sin(2.5x) * cos(-1.5x) + sin(3.5x) * cos(0.5x) = 0

Now, you could use the product-to-sum identities to further simplify this equation and try to isolate the terms involving x. The product-to-sum identities are:

sin(A) * cos(B) = 0.5 * [sin(A + B) + sin(A - B)] cos(A) * sin(B) = 0.5 * [sin(A + B) - sin(A - B)]

Applying these identities to the equation might lead to a simpler form, but it's important to note that trigonometric equations can sometimes have multiple solutions or be quite complex to solve exactly. You might consider using numerical methods or graphing software to visualize the solutions if an exact analytical solution is not immediately apparent.