i have some problems doing trig
the first one is: Show that cos(x/2) sin(3x/2) = ½(sinx + sin2x)
i know that you are supposed to substitute all those trig function things in it but i kind of forgot how to
the only that i can see substituting in is the double angle one for sin2x
could anyone walk me through the process maybe?
What a very nasty trig identity.
I am sure this is not the most efficient way, but the neat thing about identities, if you do legitimate steps, usually you end up showing LS = RS.
I end up with a lot of x/2 angles, so whenever I have one of those I will replace it with A
LS= cos(x/2)sin(3x/2)
=cosA(sin(x+x/2))
=cosA(sinxcos(x/2) + cosxsin(x/2))
=cosA(2sinAcosAcosA + (cos^2 A - sin^2 A)sinA)
= 2(sin^2 A)(cos^3 A + sinA(cos^3 A) - (sin^3 A)(cosA)
=(sinA(cosA)[(3cos^2 A) - (sin^2 A)]
let that one sit for a while
R.S.
= 1/2(2sinAcosA + 2sinxcosx_
=sinAcosA + sinxcosx
=sinAcosA + (2sinAcosA)(cosx)
=sinAcosA(1 + 2cosx)
=sinAcosA(1 + cosx + cosx)
=sinAcosA(sin^2A + cos^2A + 2cos^2A -1 + 1 - 2sin^2A)
= sinAcosA(3cos^2A - sin^2A)
= L.S. !!!!!!!!!
please, somebody come up with a better way.
To prove the trigonometric identity cos(x/2) sin(3x/2) = ½(sinx + sin2x), you need to manipulate the left side of the equation so that it equals the right side.
Here's a step-by-step process:
1. Start with the left side of the equation: cos(x/2) sin(3x/2).
2. Apply the double angle formula for sin(2x): sin(2x) = 2sin(x)cos(x).
3. Substitute sin(3x/2) with sin(2x + x/2). This is because sin(3x/2) can be expressed as sin(2x + x/2), using the sum of angles formula for sine.
4. Expand sin(2x + x/2) using the sum of angles formula for sine: sin(2x + x/2) = sin(2x)cos(x/2) + cos(2x)sin(x/2).
5. Replace sin(2x) with 2sin(x)cos(x) (from the double angle formula).
6. Substitute cos(2x) with cos^2(x) - sin^2(x) (from the double angle formula).
7. Simplify the expression using the trigonometric identity sin^2(x) + cos^2(x) = 1.
8. Simplify further by factoring out common terms.
9. The resulting expression on the left side should be equal to the right side, which is ½(sinx + sin2x).