Question

Решить уравнение sgrt(2)sin(2x+pi /4 )-sgrt(2)*sinx=sin2x+1

Answer 1

√2 * ( sin 2x * cos pi/4  + cos 2x * sin pi/4)  + √ 2 * cos x = sin 2x - 1;√2 *  (sin 2x * √2 / 2  + cos 2x * √2/2)  + √2 * cos x = sin 2x - 1; sin 2x + cos 2x + √2* cos x = sin 2x - 1; cjs 2x + √2 cos x + 1= 0;2 cos^2 x - 1 + √2 cos x + 1= 0;2 cos^2 x + √2 cos x = 0; cos x( 2 cos x + √2) = 0; cos x = 0;  x = π/2 + πk; k-Z   или         cos x = - √2 / 2; x = + - 3π/4 + 2 πn ; n-Z.часть б) - 5π/2; - 3π/2; - 5π/4.

Answer 2

To solve the equation √2sin(2x + π/4) - √2sin(x) = sin(2x) + 1, we'll follow these steps:

Step 1: Simplify the equation using trigonometric identities. Step 2: Solve the simplified equation.

Let's get started:

Step 1: Simplify the equation First, we'll simplify the equation using the following trigonometric identities:

1. sin(2x) = 2sin(x)cos(x)
2. sin(π/4) = cos(π/4) = 1/√2

Now, the equation becomes: √2sin(2x + π/4) - √2sin(x) = 2sin(x)cos(x) + 1

Step 2: Solve the equation Let's put everything on one side of the equation:

√2sin(2x + π/4) - √2sin(x) - 2sin(x)cos(x) - 1 = 0

Now, we have a nonlinear trigonometric equation, and solving it analytically can be quite challenging. However, we can use numerical methods or graphs to approximate the solutions.

To get an approximate solution, let's graph the function:

f(x) = √2sin(2x + π/4) - √2sin(x) - 2sin(x)cos(x) - 1

Using a graphing calculator or a graphing tool, you can plot this function and find the approximate solutions by identifying the x-values where f(x) = 0.

Keep in mind that there may be multiple solutions depending on the range of x you are interested in. Numerical methods like the Newton-Raphson method or the bisection method can also be used to find more accurate solutions.

Please note that since this is a complex equation, the exact solutions may not be easily expressible in simple closed-form expressions. The approach mentioned above will help you find approximate numerical solutions.