Question

Sin^2(2x) + sin^2(4x) = 1

Answer 1

Ответ:

x={π/4+kπ/2; ±π/12+kπ/2}, k∈Z

Объяснение:

sin²2x + sin²4x = 1

Формула: sin²x=(1-cos2x)/2; sin²x+cos²x=1

sin²2x=(1-cos4x)/2; sin²4x=1-cos²4x

(1-cos4x)/2+(1-cos²4x)=1

1-cos4x+2-2cos²4x=2

2cos²4x+cos4x-1=0

cos4x=t⇒|t|≤1

2t²+t-1=0

D=1-4·2·(-1)=9=3²

t₁=(-1-3)/4=-1

t₂=(-1+3)/4=0,5

1) cos4x=-1

4x=π+2kπ

x=π/4+kπ/2, k∈Z

2) cos4x=0,5

4x=±π/3+2kπ

x=±π/12+kπ/2, k∈Z

Answer 2

To solve the equation sin^2(2x) + sin^2(4x) = 1, we can use trigonometric identities and algebraic manipulations. Let's break it down step by step:

1. Start with the trigonometric identity: sin^2(theta) + cos^2(theta) = 1. This is the Pythagorean identity, which holds true for any angle theta.

2. We can rewrite the given equation as follows: sin^2(2x) + sin^2(4x) = sin^2(2x) + (1 - cos^2(4x)) = 1.

3. Now, let's use another trigonometric identity: cos(2theta) = 1 - 2sin^2(theta). Rearranging this identity, we have sin^2(theta) = (1 - cos(2theta))/2.

4. Applying the identity to the equation, we get: sin^2(2x) + (1 - cos^2(4x)) = sin^2(2x) + (1 - cos(8x))/2 = 1.

5. Multiplying the equation by 2 to eliminate the denominator, we have: 2sin^2(2x) + 1 - cos(8x) = 2.

6. Rearranging terms, we obtain: 2sin^2(2x) - cos(8x) = 1.

At this point, the equation is still complex and nonlinear, and it's not straightforward to find the exact solutions analytically. However, numerical methods or approximation techniques can be used to find approximate solutions.