Question

2cos^2(2х+пи/3)-sin^2(х+пи/6)=2

Answer 1

Ответ:

Пошаговое объяснение:

2cos²(2x + π/3) - sin²(x + π/6) = 2

2cos²(2x + π/3) - (1 - cos(2x + π/3))/2 = 2

4cos²(2x + π/3) - 1 + cos(2x + π/3) = 4

4cos²(2x + π/3) + cos(2x + π/3) - 5 = 0

Замена: t = cos(2x + π/3), |t| ≤ 1.

4t² + t - 5 = 0

D = 81

t₁ = (-1 - 9)/(2·4) = -5/4 - не удовлетворяет условию |t| ≤ 1

t₂ = (-1 + 9)/(2·4) = 1

cos(2x + π/3) = 1

2x + π/3 = 2πn, n ∈ Z

x = -π/6 + πn, n ∈ Z

Answer 2

To solve the trigonometric equation 2cos^2(2x+π/3) - sin^2(x+π/6) = 2, we'll use trigonometric identities and algebraic manipulation. Let's go step by step:

Step 1: Use the double angle formula for cosine. cos(2θ) = 2cos^2(θ) - 1

Step 2: Use the sum-to-product formula for sine. sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

Step 3: Apply the formulas in the equation and simplify. 2[2cos^2(x + π/3) - 1] - [sin(x)cos(π/6) + cos(x)sin(π/6)]^2 = 2

Step 4: Simplify further using the value of cos(π/6) and sin(π/6). cos(π/6) = √3/2 sin(π/6) = 1/2

Now, let's apply these values: 2[2cos^2(x + π/3) - 1] - [sin(x) * (√3/2) + cos(x) * (1/2)]^2 = 2

Step 5: Expand and simplify. 2[2cos^2(x + π/3) - 1] - [√3/2 * sin(x) + 1/2 * cos(x)]^2 = 2

Step 6: Expand the square term. 2[2cos^2(x + π/3) - 1] - [(√3/2)^2 * sin^2(x) + 2 * (√3/2) * (1/2) * sin(x)cos(x) + (1/2)^2 * cos^2(x)] = 2

Step 7: Simplify further using trigonometric identities. 2[2cos^2(x + π/3) - 1] - [(3/4) * sin^2(x) + (√3/2) * sin(x)cos(x) + (1/4) * cos^2(x)] = 2

Step 8: Distribute the 2 in the first term. 4cos^2(x + π/3) - 2 - (3/4) * sin^2(x) - (√3/2) * sin(x)cos(x) - (1/4) * cos^2(x) = 2

Step 9: Move all terms to one side of the equation. 4cos^2(x + π/3) - (3/4) * sin^2(x) - (√3/2) * sin(x)cos(x) - (1/4) * cos^2(x) = 4

Step 10: Combine the cosine and sine terms using the identity sin^2(θ) + cos^2(θ) = 1. 4cos^2(x + π/3) - (3/4) * (1 - cos^2(x)) - (√3/2) * sin(x)cos(x) - (1/4) * cos^2(x) = 4

Step 11: Expand the cosine terms. 4cos^2(x + π/3) - (3/4) + (3/4) * cos^2(x) - (√3/2) * sin(x)cos(x) - (1/4) * cos^2(x) = 4

Step 12: Combine the cos^2(x) terms and move constants to the other side. 4cos^2(x + π/3) + (3/4) * cos^2(x) - (√3/2) * sin(x)cos(x) = 4 + 3/4

Step 13: Factor out the common factor of cos^2(x). cos^2(x + π/3) + (3/4) * cos^2(x) - (√3/2) * sin(x)cos(x) = 4 + 3/4

Step 14: Now, apply the double angle formula for cosine (cos(2θ) = 2cos^2(θ) - 1) to the first term on the left side. 2cos^2(x + π/3) - 1 + (3/4) * cos^2(x) - (√3/2) * sin(x)cos(x) = 4 + 3/4

Step 15: Combine the cos^2(x) terms. 2cos^2(x + π/3) + (3/4) * cos^2(x) - 1 - (√3/2) * sin(x)cos(x) = 4 + 3/4

Step 16: Move the constant term to the right side. 2cos^2(x + π/3) + (3/4) * cos^2(x) - (√3/2) * sin(x)cos(x) = 4 + 3/4 + 1

Step 17: Combine the constant terms on the right side. 2cos^2(x + π/3) + (3/4) * cos^2(x) - (√3/2) * sin(x)cos(x) = 4 + 4/4

Step 18: Simplify the right side. 2cos^2(x + π/3) + (3/4) * cos^2(x) - (√3/2) * sin(x)cos(x) = 5

Now, the equation is in a manageable form. Solving this equation may involve further trigonometric identities and techniques, but at this point, it's better suited for numerical methods or graphing calculators to find approximate solutions.