Simplify:
(cosx + cos2x + cos3x + cos4x + cos5x + cos6x) / (sinx + sin2x + sin3x + sin4x + sin5x + sin6x)
into a single Cotangent function.
Using the sum-to-products, I was able to get remove some of the addition in attempts to get full multiplication, but I could not get the 2 out of the denominator for the 5x or 6x.
Am I going about this the wrong way? Any ideas?
Here's the whole deal:
Let f(x) = (cos(x)+cos(2x)+...cos(6x)) / (sin(x)+...+sin(6x))
Multiply the denominator by 2sin(x/2) to get six products of 2sin(x/2)*sin(nx) where n=1 to 6.
Using the sum and product formula, you should get:
denominator =
cos(x/2)-cos(3x/2)+
cos(3x/2)-cos(5x/2)+
cos(5x/2)-cos(7x/2)+
cos(7x/2)-cos(9x/2)+
cos(9x/2)-cos(11x/2)+
cos(11x/2)-cos(13x/2)
= cos(x/2)-cos(13x/2)
= 2sin(7x/2)sin(3x)
Similarly, multiply the numerator by 2sin(x/2) gives
numerator = 2cos(7x/2)sin(3x)
On cancelling sin(3x), we get the simplified expression as:
cot(7x/2)
Check: for x=π/4,
cot(7x/2)=cot(7π/8)=-2.41421...
cos(π/4) + cos(2π/4) + cos(3π/4) + cos(4π/4) + cos(5π/4) + cos(6π/4)
= r+0-r-1-r+0
= -(1+sqrt(2)/2)
sin(π/4) + sin(2π/4) + sin(3π/4) + sin(4π/4) + sin(5π/4) + sin(6π/4)
= r+1+r+0-r-1
= r
= sqrt(2)/2
f(π/4)
= -(1+sqrt(2)/2) / sqrt(2)/2
= -2.41421...
Try multiplying AND dividing each series by 2sin(x/2), expand and apply the sum/product formulas.
Many of the terms should cancel and leave you something simple to work with.
To simplify the given expression and write it as a single Cotangent function, we need to apply some trigonometric identities.
First, let's rewrite the expression:
(cosx + cos2x + cos3x + cos4x + cos5x + cos6x) / (sinx + sin2x + sin3x + sin4x + sin5x + sin6x)
To simplify this expression, we can use the sum-to-product trigonometric identities. These identities state that:
1) cos(a) + cos(b) = 2cos((a+b)/2)cos((a-b)/2)
2) sin(a) + sin(b) = 2sin((a+b)/2)cos((a-b)/2)
Now, let's simplify the numerator using the sum-to-product identity:
cosx + cos2x = 2cos(3x/2)cos(x/2)
Similarly, we can simplify the other terms in the numerator:
cos3x + cos4x = 2cos(7x/2)cos(x/2)
cos5x + cos6x = 2cos(11x/2)cos(x/2)
Now let's simplify the denominator using the sum-to-product identity:
sinx + sin2x = 2sin(3x/2)cos(x/2)
Similarly, we can simplify the other terms in the denominator:
sin3x + sin4x = 2sin(7x/2)cos(x/2)
sin5x + sin6x = 2sin(11x/2)cos(x/2)
Now, substitute these simplified expressions back into the original expression:
(2cos(3x/2)cos(x/2) + 2cos(7x/2)cos(x/2) + 2cos(11x/2)cos(x/2)) / (2sin(3x/2)cos(x/2) + 2sin(7x/2)cos(x/2) + 2sin(11x/2)cos(x/2))
Factor out the common factor of 2cos(x/2) from the numerator and denominator:
2cos(x/2)(cos(3x/2) + cos(7x/2) + cos(11x/2)) / 2cos(x/2)(sin(3x/2) + sin(7x/2) + sin(11x/2))
Cancel out the common factors:
cos(3x/2) + cos(7x/2) + cos(11x/2) / sin(3x/2) + sin(7x/2) + sin(11x/2)
Finally, we can simplify this expression further by using the Cotangent identity:
cot(3x/2) = cos(3x/2) / sin(3x/2)
cot(7x/2) = cos(7x/2) / sin(7x/2)
cot(11x/2) = cos(11x/2) / sin(11x/2)
Therefore, the simplified expression can be written as:
cot(3x/2) + cot(7x/2) + cot(11x/2)
To simplify the expression
(cosx + cos2x + cos3x + cos4x + cos5x + cos6x) / (sinx + sin2x + sin3x + sin4x + sin5x + sin6x)
into a single cotangent function, you can make use of the sum-to-product identities and the cotangent identity.
Let's start by applying the sum-to-product identities to the numerator and denominator separately. The sum-to-product identities state:
cos(A) + cos(B) = 2 * cos((A+B)/2) * cos((A-B)/2)
sin(A) + sin(B) = 2 * sin((A+B)/2) * cos((A-B)/2)
Using these identities, we can rewrite the denominator:
sinx + sin2x + sin3x + sin4x + sin5x + sin6x
= 2 * (sin((x+2x)/2) * cos((2x-x)/2) + sin((2x+4x)/2) * cos((4x-2x)/2) + sin((4x+6x)/2) * cos((6x-4x)/2))
= 2 * (sin(3x/2) * cos(x/2) + sin(3x) * cos(x) + sin(5x/2) * cos(2x))
Similarly, we can rewrite the numerator using the sum-to-product identity:
cosx + cos2x + cos3x + cos4x + cos5x + cos6x
= 2 * (cos((x+2x)/2) * cos((2x-x)/2) + cos((2x+4x)/2) * cos((4x-2x)/2) + cos((4x+6x)/2) * cos((6x-4x)/2))
= 2 * (cos(3x/2) * cos(x/2) + cos(3x) * cos(x) + cos(5x/2) * cos(2x))
Now, we have:
(2 * (cos(3x/2) * cos(x/2) + cos(3x) * cos(x) + cos(5x/2) * cos(2x))) /
(2 * (sin(3x/2) * cos(x/2) + sin(3x) * cos(x) + sin(5x/2) * cos(2x)))
The 2 in both the numerator and denominator cancels out, leaving:
(cos(3x/2) * cos(x/2) + cos(3x) * cos(x) + cos(5x/2) * cos(2x)) /
(sin(3x/2) * cos(x/2) + sin(3x) * cos(x) + sin(5x/2) * cos(2x))
Now, we can use the cotangent identity:
cot(A) = cos(A) / sin(A)
Applying this identity, we can rewrite the expression as:
cot(3x/2) * cot(x/2) + cot(3x) * cot(x) + cot(5x/2) * cot(2x)
Thus, the simplified expression is:
cot(3x/2) * cot(x/2) + cot(3x) * cot(x) + cot(5x/2) * cot(2x)