Question

Sin a +2sin a2 +sin3a = 4sin2a×cos^2 a/2решите уравнение​

Answer 1

ответ выйдет ну чесно говоря sin a+2sina2+sin3a =4sin2a×cos^2a/2 ответ sin1/2 квадрат

Answer 2

To solve the equation, we first need to simplify the expression on the right-hand side and then set it equal to the left-hand side. Here's the step-by-step solution:

Given equation: sin(a) + 2sin(a^2) + sin(3a) = 4sin^2(a) * cos^2(a/2)

Step 1: Simplify the right-hand side

We know the following trigonometric identities:

• sin(2θ) = 2sin(θ) * cos(θ)
• cos^2(θ) = 1 - sin^2(θ)

Let's use these identities to simplify the right-hand side of the equation:

4sin^2(a) * cos^2(a/2) = 4sin^2(a) * (1 - sin^2(a/2)) = 4sin^2(a) - 4sin^2(a) * sin^2(a/2)

Step 2: Rewrite sin(3a) using the identity for sin(3θ)

sin(3θ) = 3sin(θ) - 4sin^3(θ)

So, sin(3a) = 3sin(a) - 4sin^3(a)

Now, the equation becomes:

sin(a) + 2sin(a^2) + (3sin(a) - 4sin^3(a)) = 4sin^2(a) - 4sin^2(a) * sin^2(a/2)

Step 3: Combine like terms and rearrange the equation

Group the terms with sin(a) and move everything to the left-hand side:

3sin(a) - 4sin^3(a) + sin(a) - 4sin^2(a) + 2sin(a^2) + 4sin^2(a) * sin^2(a/2) = 0

Combine the terms with sin(a):

4sin(a) - 4sin^3(a) + 2sin(a^2) + 4sin^2(a) * sin^2(a/2) = 0

Step 4: Factor out common terms

Factor out sin(a) from the first two terms and 2sin(a^2) from the last two terms:

4sin(a)(1 - sin^2(a)) + 2sin(a^2)(1 + 2sin^2(a/2)) = 0

Step 5: Use the identity sin^2(a) = 1 - cos^2(a)

Replace sin^2(a) in the first term with 1 - cos^2(a):

4sin(a)(1 - (1 - cos^2(a))) + 2sin(a^2)(1 + 2sin^2(a/2)) = 0

Simplify further:

4sin(a) * cos^2(a) + 2sin(a^2)(1 + 2sin^2(a/2)) = 0

Step 6: Solve for sin(a^2)

We can now solve for sin(a^2):

2sin(a^2)(1 + 2sin^2(a/2)) = -4sin(a) * cos^2(a)

Divide both sides by 2(1 + 2sin^2(a/2)):

sin(a^2) = (-4sin(a) * cos^2(a)) / (2(1 + 2sin^2(a/2)))

sin(a^2) = -2sin(a) * cos^2(a) / (1 + 2sin^2(a/2))

Step 7: Use the double-angle identity for sin(2θ)

sin(2θ) = 2sin(θ) * cos(θ)

Substitute a^2 with 2a:

sin(2a) = 2sin(a) * cos(a)

Step 8: Solve for sin(a/2)

Now, we can solve for sin(a/2):

sin(a/2) = ±√((1 - cos(a))/2)

Step 9: Determine the possible solutions for a

At this point, we have obtained sin(a/2) = ±√((1 - cos(a))/2), and there are multiple possible values of a that satisfy this equation. However, without a specific value or range for a, we cannot provide the exact solutions.

To find specific solutions for a, you would need additional information or constraints on the value of a. If you have a specific value of a or any other constraints, please provide them, and I can help you find the solutions accordingly.