Question

Sin(x-pi/4)+cos(x-pi/4)=sin2x

Answer 1

В левой части используем формулу дополнительного угла (об этом можете найти в интернете или в книжке).

$\sqrt{1^2+1^2}\sin\left(x-\frac{\pi}{4}+\arcsin\frac{1}{\sqrt{1^2+1^2}}\right)=\sin 2x\\ \\ \sqrt{2}\sin\left(x-\frac{\pi}{4}+\frac{\pi}{4}\right)=\sin 2x\\ \\ \sqrt{2}\sin x-2\sin x\cos x=0\\ \\ \sin x\left(\sqrt{2}-2\cos x\right)=0\\ \\ \sin x=0~~~~\Longrightarrow~~~ \boxed{x_1=\pi k,k \in Z}\\ \\ \cos x=\frac{\sqrt{2}}{2}~~~\Longrightarrow~~~ \boxed{x_2=\pm\frac{\pi}{4}+2\pi n,n \in Z}$

Answer 2

To solve the trigonometric equation sin(x - π/4) + cos(x - π/4) = sin(2x), we can use trigonometric identities and algebraic manipulations to simplify the equation and find its solutions.

Let's break down the equation step by step:

1. Start with the given equation: sin(x - π/4) + cos(x - π/4) = sin(2x)

2. Apply the angle sum identity for sine and cosine: sin(x)cos(π/4) - cos(x)sin(π/4) + cos(x)cos(π/4) + sin(x)sin(π/4) = sin(2x)

3. Simplify the trigonometric terms: (1/√2)sin(x) - (1/√2)cos(x) + (1/√2)cos(x) + (1/√2)sin(x) = sin(2x)

4. Combine like terms: (1/√2)sin(x) + (1/√2)sin(x) = sin(2x) (2/√2)sin(x) = sin(2x)

5. Simplify the left-hand side: √2 * sin(x) = sin(2x)

6. Apply the double-angle identity for sine: √2 * sin(x) = 2 * sin(x) * cos(x)

7. Divide both sides by sin(x) (assuming sin(x) ≠ 0): √2 = 2 * cos(x)

8. Solve for cos(x): cos(x) = √2 / 2

At this point, you can use the fact that the cosine value is positive in the first and fourth quadrants, so the solutions for x in the interval [0, 2π) are: x = π/4 and x = 7π/4.

These are the solutions to the given trigonometric equation sin(x - π/4) + cos(x - π/4) = sin(2x).