Prove That: sin(x/2)sin(7x/2) + sin(3x/2)sin(11x/2) = sin(2x)sin(5x)?

Question

i got this math worksheet and i really can't figure out how to do this one, can anyone figure out which formula to use?

Answer 1

1. To prove the equation, start with the left side of the equation.

2. Use this trigonometric identity:

sin(x) + sin (y) = 1/2 ( cos(x-y) - cos (x+y) )

for each sin (x/2)sin(7x/2) and sin(3x/2)sin(11x/2).
(You'll get [cos (3x) - cos (7x)]/2)

3. then, use this trigonometric identity:

cos (x) - cos(y) = -2 * sin[(x+y)/2] * sin [(x-y)/2]

GOOD LUCK! =)

Answer 2

This one seems pretty tedious. I think you'll first have to use the product to sum formula for the sin(x/2)sin(7x/2) and sin(3x/2)sin(11x/2). That identity is:

sin u * sin v = (1/2)[cos(u - v) - cos(u + v)]

And I think you'll have to do some algebraic manipulation...and then probably have to use a sum to product formula since you would have the sin terms adding. That identity is:

sin u + sin v = 2sin[(u+v)/2] cos[(u-v)/2]

And I'm sure there's a few other identities and quite a bit of algebraic manipulation to use. I know its not much help, but maybe its some place to start.

Good luck!

Answer 3

Let x/2 = y for simlicity
sin(x/2) sin (7x/2) = siny sin 7y

sin A sin B = (Cos(A-B) - Cos(A+B))/2

so 1st term is (Cos6y - cos 8y)/2
2nd term is (Cos7y - Cos 4y)/2

Add the 2 and simplify

Answer 4

here is what i can tell you

sin(x/2) = sqrt((1 - cosx)/2)
sin((7x)/2) = sqrt((1 - cos(7x))/2)
sin((3x)/2) = sqrt((1 - cos(3x))/2)
sin((11x)/2) = sqrt((1 - cos(11x))/2)

sin(2x) = 2sinxcosx
sin(5x) = (sinx)^5 - 10(cosx)^2(sinx)^3 + 5(cosx)^4sinx

Answer 5

sin(x / 2) · sin(7x / 2) + sin(3x / 2) · sin(11x / 2) = sin(2x) · sin(5x)

Use sin(a) · sin(b) = [cos(a - b) - cos(a + b)] / 2.

Left addend: sin(x / 2) · sin(7x / 2)
[cos(x / 2 - 7x / 2) - cos(x / 2 + 7x / 2)] / 2
[cos(-3x) - cos(4x)] / 2

Right addend: sin(3x / 2) · sin(11x / 2)
[cos(3x / 2 - 11x / 2) - cos(3x / 2 + 11x / 2)] / 2
[cos(-4x) - cos(7x)] / 2

Sum: sin(2x) · sin(5x)
[cos(2x - 5x) - cos(2x + 5x)] / 2
[cos(-3x) - cos(7x)] / 2

This leaves you
[cos(-3x) - cos(4x)] / 2 + [cos(-4x) - cos(7x)] / 2 = [cos(-3x) - cos(7x)] / 2
Mulitply by 2:
cos(-3x) - cos(4x) + cos(-4x) - cos(7x) = cos(-3x) - cos(7x)
Negative-angle formulas:
cos(3x) - cos(4x) + cos(4x) - cos(7x) = cos(3x) - cos(7x)
Combine like terms:
cos(3x) - cos(7x) = cos(3x) - cos(7x)
True statement... done deal!

Answer 6

No, you prove it.

Answer 7

Who'd want to to THAT?
:))))))))))

Answer 8

no.