Question

а) 2sinx+ √ 2=0. б) sin3XcosX + cos3XsinX=0 в) cos6x = cos4x P.s. Я знаю что это элементарно, но правильно оформить и решить не могу....

Answer 1

a) 2sin(x) + √2 = 0

To solve the equation 2sin(x) + √2 = 0, we need to isolate the variable sin(x). Here's how we can do it:

1. Subtract √2 from both sides of the equation: 2sin(x) = -√2

2. Divide both sides of the equation by 2: sin(x) = -√2/2

Now, we need to find the values of x that satisfy this equation. To do that, we can use the inverse sine function (also known as arcsin or sin^-1) to find the angle whose sine is equal to -√2/2.

Using a calculator or a trigonometric table, we find that the angle whose sine is -√2/2 is -π/4 or -45 degrees. However, keep in mind that the sine function is periodic, so there are infinitely many solutions to this equation.

The general solution for sin(x) = -√2/2 is: x = -π/4 + 2πn, where n is an integer.

This means that any angle that is -π/4 radians (or -45 degrees) plus a multiple of 2π radians (or 360 degrees) will satisfy the equation.

b) sin(3x)cos(x) + cos(3x)sin(x) = 0

To solve the equation sin(3x)cos(x) + cos(3x)sin(x) = 0, we can use the trigonometric identity for the sine of the sum of two angles:

sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

By comparing the given equation with the trigonometric identity, we can see that A = 3x and B = x. Therefore, we have:

sin(3x + x) = 0

Simplifying the equation, we get:

sin(4x) = 0

To find the values of x that satisfy this equation, we need to find the angles whose sine is equal to 0. The sine function is equal to 0 at multiples of π (or 180 degrees). Therefore, the solutions to this equation are:

4x = nπ, where n is an integer.

Dividing both sides of the equation by 4, we get:

x = nπ/4, where n is an integer.

This means that any angle that is a multiple of π/4 radians (or 45 degrees) will satisfy the equation.

c) cos(6x) = cos(4x)

To solve the equation cos(6x) = cos(4x), we can use the cosine of the difference of two angles identity:

cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

By comparing the given equation with the trigonometric identity, we can see that A = 6x and B = 4x. Therefore, we have:

cos(6x - 4x) = cos(6x)cos(4x) + sin(6x)sin(4x)

Simplifying the equation, we get:

cos(2x) = cos(6x)cos(4x) + sin(6x)sin(4x)

Since the cosine function is an even function, we can rewrite the equation as:

cos(2x) = cos(6x)cos(4x) - sin(6x)sin(4x)

Now, we can use the cosine of the sum of two angles identity:

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

By comparing the equation with the trigonometric identity, we can see that A = 2x and B = 6x - 4x. Therefore, we have:

cos(2x + 2x) = cos(2x)cos(6x - 4x) - sin(2x)sin(6x - 4x)

Simplifying the equation, we get:

cos(4x) = cos(2x)cos(2x) - sin(2x)sin(2x)

Using the double-angle identity for cosine:

cos(4x) = cos^2(2x) - sin^2(2x)

Now, we can substitute cos^2(2x) with 1 - sin^2(2x) (using the Pythagorean identity for sine and cosine):

cos(4x) = 1 - sin^2(2x) - sin^2(2x)

Simplifying the equation, we get:

cos(4x) = 1 - 2sin^2(2x)

Rearranging the equation, we have:

2sin^2(2x) + cos(4x) - 1 = 0

Unfortunately, this equation cannot be easily solved algebraically. To find the solutions, we would need to use numerical methods or a graphing calculator.

I hope this helps! Let me know if you have any further questions.