So I have two questions that have been puzzling me for quite some time and would really appreciate any help with either of them!
(a) There are four positive intergers a, b, c, and d such that 4cos(x)cos(2x)cos(4x)=cos(ax)+cos(bx)+cos(cx)+cos(dx) for all values of x. Find a+b+c+d.
I started this problem by trying to use the double-angle formulas to expand cos(2x) and cos(4x). This quickly seemed to become difficult as I was working with exponents and there didn't seem to be an easy way to simplify
(b) There are intergers a, b, c, and d such that tan(7.5)=sqrt(a)+sqrt(b)-sqrt(c)-sqrt(d). Find a+b+c+d.
I started this problem by using the fact that tan(x)=sin(x)/cos(x). Next I used the half-angle identities and sum-difference formulas to get a very ugly fraction that I had no idea how to solve.
Any help would be much appreciated - thank you in advance!
(a) Try using the product-to-sum formula
cos(a)cos(b) = 1/2 (cos(a-b)+cos(a+b))
(b) We have
tan x/2 = (1-cosx)/sinx
tan 15 = (1-cos30)/sin30 = (1-√3/2)/(1/2) = 2-√3
sin15 = √((1-cos30)/2) = √((1-√3/2)/2) = √(2-√3)/2
sin15 = √((1+cos30)/2) = √((1+√3/2)/2) = √(2+√3)/2
Now apply that again, and note that
(√2+√6)^2 = 4(2+√3)
and I think things will fall out as you desire.
Thank you so much for your help! I got both of those now!
For question (a), you have a trigonometric expression on the left-hand side and a sum of cosine functions on the right-hand side. To find the values of a, b, c, and d, you can compare the coefficients of cosine functions on both sides of the equation.
Start by expanding the left-hand side using the double-angle formula for cosine:
4cos(x)cos(2x)cos(4x) = 4cos(x)(2cos^2(2x) - 1)cos(4x)
= 8cos(x)cos^2(2x)cos(4x) - 4cos(x)cos(4x)
Next, we can simplify further by using the product-to-sum formulas and the double-angle formula again:
8cos(x)cos^2(2x)cos(4x) - 4cos(x)cos(4x)
= 8(1/2)(cos(3x) + cos(-3x))cos(4x) - 4cos(4x)
= 4cos(4x)cos(3x) + 4cos(-4x)cos(3x) - 4cos(4x)
= 4cos(4x)[cos(3x) - 1]
Now we have an expression of the form 4cos(4x)[cos(3x) - 1]. Let's compare this to the right-hand side of the equation, cos(ax) + cos(bx) + cos(cx) + cos(dx).
From the expression on the left-hand side, we can see that a = 4 and d = 4 because they correspond to the coefficients of cos(4x).
Now, compare the remaining term, 4cos(4x)(cos(3x) - 1), to the sum of cosine functions on the right-hand side. By comparing the coefficients of cos(3x), we can equate:
4cos(4x)(cos(3x) - 1) = cos(bx) + cos(cx)
Since the right-hand side contains only two cosine functions, we can conclude that b = 3 and c = 3. Therefore, a + b + c + d = 4 + 3 + 3 + 4 = 14.
For question (b), we have the equation tan(7.5) = sqrt(a) + sqrt(b) - sqrt(c) - sqrt(d). We want to find the values of a, b, c, and d.
Using the identity tan(x) = sin(x)/cos(x), we can rewrite the equation as:
sin(7.5)/cos(7.5) = sqrt(a) + sqrt(b) - sqrt(c) - sqrt(d)
Let's rationalize the denominator by multiplying both the numerator and denominator by cos(7.5):
sin(7.5) * cos(7.5) / [cos(7.5) * cos(7.5)] =
[sin(7.5) * cos(7.5)] / [cos^2(7.5)]
Now, we can use the half-angle identity for sin(x) to simplify sin(7.5) * cos(7.5):
sin(7.5) * cos(7.5) = (1/2)sin(15) = (1/2)sin(2 * 7.5)
Applying the double-angle formula for sine, sin(2 * 7.5) = 2sin(7.5)cos(7.5), we have:
(1/2)sin(2 * 7.5) = (1/2) * 2sin(7.5)cos(7.5) =
sin(7.5)cos(7.5)
Therefore, the equation becomes:
[sin(7.5)cos(7.5)] / [cos^2(7.5)] =
[sqrt(a) + sqrt(b) - sqrt(c) - sqrt(d)] / [1]
Since both sides of the equation are equal, the terms inside the square brackets must be equal. This gives us the following system of equations:
sqrt(a) + sqrt(b) - sqrt(c) - sqrt(d) = sin(7.5)cos(7.5) (1)
cos^2(7.5) = 1 (2)
From equation (2), we know that cos^2(7.5) = 1. Solving for cos(7.5), we get cos(7.5) = ±1.
Now, we need to consider the sign of cos(7.5) to determine the signs of the square roots in equation (1). Since tan(7.5) is positive, we know that cos(7.5) must be positive as well.
Therefore, cos(7.5) = 1, and the equation (1) becomes:
sqrt(a) + sqrt(b) - sqrt(c) - sqrt(d) = sin(7.5)
Now, square both sides of the equation to eliminate the square roots:
[a + 2sqrt(ab) + b - 2sqrt(ac) - 2sqrt(ad) + 2sqrt(bc) - 2sqrt(bd) + c + 2sqrt(cd) - 2sqrt(d)] = sin^2(7.5)
By comparing the coefficients of the square roots, we can match the terms on both sides.
In this case, it's difficult to determine the exact values of a, b, c, and d without further information or simplification of the equation. However, we can use the equation above to isolate each term with a square root on one side and solve for them individually.
Once you have the values of a, b, c, and d, you can find their sum: a + b + c + d.
Note: The solution for question (b) involves more complex calculations, and it may not be readily solvable without additional information or further simplification. It's important to check if there are any additional relationships or trigonometric identities that could potentially simplify the equation or provide insights into finding the solution.