7cos (6п/5-x)+5cos (x-6п/5)=1

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вопрос

To solve this equation, we can use the trigonometric identity:

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

Let's rewrite the equation using this identity:

7(cos(6π/5)cos(x) + sin(6π/5)sin(x)) + 5(cos(x)cos(6π/5) - sin(x)sin(6π/5)) = 1

Now, simplify the equation:

7cos(6π/5)cos(x) + 7sin(6π/5)sin(x) + 5cos(x)cos(6π/5) - 5sin(x)sin(6π/5) = 1

Now, we need to express sin(6π/5) and cos(6π/5) in terms of sin(x) and cos(x) using the trigonometric identities sin(a) = cos(a - π/2) and cos(a) = sin(a + π/2):

sin(6π/5) = cos(6π/5 - π/2) = cos(6π/5 - 3π/10) = cos(6π/10 - 3π/10) = cos(3π/10) = sin(π/2 - 3π/10) = sin(π/2)cos(3π/10) - cos(π/2)sin(3π/10) = cos(3π/10)

cos(6π/5) = sin(6π/5 + π/2) = sin(6π/5 + 3π/10) = sin(6π/10 + 3π/10) = sin(3π/10) = sin(π/2 - 3π/10) = cos(π/2)cos(3π/10) + sin(π/2)sin(3π/10) = -sin(3π/10)

Substitute these values back into the equation:

7(-sin(3π/10)cos(x) + cos(3π/10)sin(x)) + 5(cos(x)(-sin(3π/10)) - sin(x)sin(3π/10)) = 1

Now, simplify the equation further:

-7sin(3π/10)cos(x) + 7cos(3π/10)sin(x) - 5sin(3π/10)cos(x) - 5sin(x)sin(3π/10) = 1

Combine terms:

(-7sin(3π/10) - 5sin(3π/10))cos(x) + (7cos(3π/10) - 5sin(3π/10))sin(x) = 1

Simplify further:

-12sin(3π/10)cos(x) + 7cos(3π/10)sin(x) = 1

At this point, we can see that the equation does not simplify further. Therefore, the solution to the original equation is:

-12sin(3π/10)cos(x) + 7cos(3π/10)sin(x) = 1