Question

20cos3x+12cos5x-15cos4x

Answer 1

20cos3x+12cos5x-15cos4x
P1=2п/3
P2=2п/5
P3=п/2
ответ - P=2п

Answer 2

To simplify the given expression, 20cos(3x) + 12cos(5x) - 15cos(4x), we can use trigonometric identities to rewrite the terms.

1. Cosine triple angle identity: cos(3x) = 4cos^3(x) - 3cos(x)
2. Cosine double angle identity: cos(2x) = 2cos^2(x) - 1
3. Cosine addition formula: cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

Let's start with cos(4x):

cos(4x) = cos(2x + 2x) = cos(2x)cos(2x) - sin(2x)sin(2x) = (2cos^2(x) - 1)(2cos^2(x) - 1) - (2sin(x)cos(x))(2sin(x)cos(x)) = 4cos^4(x) - 4cos^2(x) + 1 - 4sin^2(x)cos^2(x) = 4cos^4(x) - 4cos^2(x) + 1 - 4(1 - cos^2(x))cos^2(x) = 4cos^4(x) - 4cos^2(x) + 1 - 4cos^2(x) + 4cos^4(x) = 8cos^4(x) - 8cos^2(x) + 1

Next, cos(5x):

cos(5x) = cos(4x + x) = cos(4x)cos(x) - sin(4x)sin(x) = (8cos^4(x) - 8cos^2(x) + 1)cos(x) - (2sin(2x)cos(2x))sin(x) = (8cos^4(x) - 8cos^2(x) + 1)cos(x) - (2 * 2sin(x)cos(x) * cos(2x))sin(x) = (8cos^4(x) - 8cos^2(x) + 1)cos(x) - (4sin(x)cos(x) * (2cos^2(x) - 1))sin(x) = (8cos^4(x) - 8cos^2(x) + 1)cos(x) - (4sin(x)cos(x) * (2cos^2(x)) + 4sin(x)cos(x)) = (8cos^4(x) - 8cos^2(x) + 1)cos(x) - (8cos^3(x)sin(x) + 4sin(x)cos(x)) = (8cos^4(x) - 8cos^2(x) + 1)cos(x) - 4sin(x)cos(x)(2cos^2(x) + 1)

Now, let's rewrite the expression with these trigonometric identities:

20cos(3x) + 12cos(5x) - 15cos(4x)

= 20(4cos^3(x) - 3cos(x)) + 12[(8cos^4(x) - 8cos^2(x) + 1)cos(x) - 4sin(x)cos(x)(2cos^2(x) + 1)] - 15(8cos^4(x) - 8cos^2(x) + 1)

Now, distribute and simplify:

= 80cos^3(x) - 60cos(x) + 12(8cos^5(x) - 8cos^3(x) + cos(x)) - 48sin(x)cos^2(x) - 12sin(x)cos(x) - 15(8cos^4(x) - 8cos^2(x) + 1)

= 80cos^3(x) - 60cos(x) + 96cos^5(x) - 96cos^3(x) + 12cos(x) - 48sin(x)cos^2(x) - 12sin(x)cos(x) - 120cos^4(x) + 120cos^2(x) - 15

Now, collect like terms:

= 96cos^5(x) + 80cos^3(x) - 120cos^4(x) - 60cos(x) - 48sin(x)cos^2(x) - 12sin(x)cos(x) + 12cos(x) - 15

So, the simplified expression is:

96cos^5(x) + 80cos^3(x) - 120cos^4(x) - 48sin(x)cos^2(x) - 12sin(x)cos(x) - 60cos(x) - 15