Question

Sin x+ cos x= корень 2

Answer 1

вспомним, что sinП/4=cosП/4=V2/2,   (V2 - это корень из 2)

sinx*cosП/4+cosx*sinП/4=1,   sin(x+П/4)=1,  x+П/4=П/2+2Пn,  x=П/4+2Пn,  мы домножили ур-е на V2/2

Answer 2

To solve the equation sin(x) + cos(x) = √2, we can use trigonometric identities and techniques. Let's go step by step:

1. We can use the identity sin^2(x) + cos^2(x) = 1, which holds true for all real values of x.

2. Rewrite the given equation with this identity: sin(x) + cos(x) = √2 (sin^2(x) + cos^2(x)) + 2sin(x)cos(x) = 2

3. Now, we can use the double-angle identity for sine: sin(2x) = 2sin(x)cos(x)

4. Substituting sin(2x) into the equation: 1 + sin(2x) = 2

5. Subtract 1 from both sides: sin(2x) = 1

6. Now, we need to find the angles whose sine is 1. We know that sin(π/2) = 1.

7. Set 2x equal to π/2: 2x = π/2

8. Solve for x: x = π/4

So, the solutions to the equation sin(x) + cos(x) = √2 are x = π/4 + 2πn and x = 5π/4 + 2πn, where n is an integer. These solutions represent all possible values of x that satisfy the equation.