Trigonometry

[Guest]

cos(7x)-sin(5x)=(3)^(1/2)(cos(5x)-sin(7x))

solve, pls help

[biankamiles]

Let x = t
cos(7t) − sin(5t) = (3)^(1/2)(cos(5t) − sin(7t))
cos(7t) − cos(5t) = (3)^(1/2)(sin(7t) − sin(5t))
cos(7t) − cos(5t) = 3cos((7t-5t)/2)sin((7t+5t)/2)
cos(2t) = 3cos(2t)sin(12t)
cos(2t)(1 - 3sin(12t)) = 0
2t = (n + ½)π

[Guest]

To solve the equation cos(7x) - sin(5x) = √3(cos(5x) - sin(7x)), we can simplify it using trigonometric identities. Let's break it down step by step:

Step 1: Expand the equation using the distributive property.
cos(7x) - sin(5x) = √3 * cos(5x) - √3 * sin(7x)

Step 2: Rearrange the equation by grouping like terms.
cos(7x) - √3 * cos(5x) = sin(5x) - √3 * sin(7x)

Step 3: Use the identity cos(a) - √3 * sin(a) = 2 * sin(a + π/3).
2 * sin(7x + π/3) = sin(5x) - √3 * sin(7x)

Step 4: Apply the identity sin(a) - √3 * sin(b) = 2 * sin(a - b - π/3).
2 * sin(7x + π/3) = 2 * sin(7x - 5x - π/3)

Step 5: Simplify the equation further.
sin(7x + π/3) = sin(2x - π/3)

Step 6: Use the identity sin(a) = sin(b) if a = b + 2nπ or a = π - b + 2nπ, where n is an integer.
7x + π/3 = 2x - π/3 + 2nπ OR 7x + π/3 = π - (2x - π/3) + 2nπ

Step 7: Solve the two equations separately for x.
Equation 1: 7x + π/3 = 2x - π/3 + 2nπ
5x = -2π/3 + 2nπ - π/3
x = (-3π/3 + 2nπ) / 5

Equation 2: 7x + π/3 = π - (2x - π/3) + 2nπ
7x + π/3 = π - 2x + π/3 + 2nπ
9x = π - π/3 + 2nπ
x = (π - π/3 + 2nπ) / 9

So, the general solutions for x are:
x = (-3π/3 + 2nπ) / 5 OR x = (π - π/3 + 2nπ) / 9

Please note that 'n' represents any https://calculator-integral.com/integer value.

[Guest]

Let's substitute x=t in the given equation:
cos(7x)−sin(5x)=√(cos(5x)−sin(7x))
cos(7x)−cos(5x)=√(sin(7x)−sin(5x))
cos(7x)−cos(5x)=3cos((7x−5x)/2)sin((7x+5x)/2)
cos(2x)=3cos(2x)sin(6x)
cos(2x)(1−3sin(6x))=0
2x=(n+1/2)π

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