Question

Sinx•sin(60-x)sin•(60+x)=0,125

Answer 1

To solve the equation sin(x) * sin(60 - x) * sin(60 + x) = 0.125, we can use trigonometric identities and properties to simplify the equation and find possible solutions.

Let's break down the equation step by step:

1. We know that sin(60 - x) = sin(60)cos(x) - cos(60)sin(x) = √3/2 * cos(x) - 1/2 * sin(x).
2. Similarly, sin(60 + x) = sin(60)cos(x) + cos(60)sin(x) = √3/2 * cos(x) + 1/2 * sin(x).

Now, substitute these expressions into the original equation:

sin(x) * (sin(60)cos(x) - cos(60)sin(x)) * (sin(60)cos(x) + cos(60)sin(x)) = 0.125

Expand the equation:

sin(x) * (√3/2 * cos(x) - 1/2 * sin(x)) * (√3/2 * cos(x) + 1/2 * sin(x)) = 0.125

Now, let's simplify further:

sin(x) * (√3/2)^2 * cos(x)^2 - (1/2)^2 * sin(x)^2 = 0.125

sin(x) * 3/4 * cos(x)^2 - 1/4 * sin(x)^2 = 0.125

3/4 * sin(x) * cos(x)^2 - 1/4 * sin(x)^2 = 0.125

Now, you can factor out sin(x):

sin(x) * (3/4 * cos(x)^2 - 1/4 * sin(x)) = 0.125

Now we have two possible cases:

1. sin(x) = 0 (which gives us one set of solutions).
2. 3/4 * cos(x)^2 - 1/4 * sin(x) = 0 (which gives us another set of solutions).

For the first case, sin(x) = 0 implies x = 0, π, 2π, ... (integer multiples of π).

For the second case, we need to solve the quadratic equation:

3/4 * cos(x)^2 - 1/4 * sin(x) = 0

3 * cos(x)^2 - sin(x) = 0

3 * (1 - sin(x)^2) - sin(x) = 0

3 - 3 * sin(x)^2 - sin(x) = 0

3 * sin(x)^2 + sin(x) - 3 = 0

Now, you can solve this quadratic equation for sin(x), and then find the corresponding values of x.

Please note that solving trigonometric equations can sometimes lead to complex solutions, and this is a simplified approach. You might want to use a calculator or numerical methods to find more accurate solutions.