using sin(alpha-beta)=(sin alpha)(cos beta)-(cos alpha)(sin beta)
use the identity to derive the result
proof for : sin(alpha-beta)
PLEASE PLEASE PLEASE HELP
so you want us to use the identity:
sin(A - B) = sinAcosB - cosAsinB to prove
sin(A - B) = sinAcosB - cosAsinB ?
mmmhhh?
did you mean sin(A+B) ??
Nope, assuming I want to prove cos2theta is cos 2theta =1-2sin^2theta
cos2theta=(1-sin^2theta)-sin^2theta
cos2theta=1-2sin^2theta
That's not what you said originally, look at your post.
So which identity are we deriving ?
there are 2
sin(alpha+beta)=(sin alpha)(cos beta)+(cos alpha)(sin beta)
cos(alpha+beta)=(sis alpha)(cos beta)+(sin alpha)(sin beta)
these identities to derive
i)sin(alpha-theta)
ii)cos(alpha-theta)
iii)sin2theta
I just want one of them solved in steps to be able to implement the steps to rest of the following.
Ok, starting to make sense now
So I will assume you are given:
sin(alpha+beta)=(sin alpha)(cos beta)+(cos alpha)(sin beta)
and
cos(alpha+beta)=(sis alpha)(cos beta)+(sin alpha)(sin beta)
I will use A and B for easier typing
You must also know that
1. sin(-A) = -sinA
2. cos(-A) = cosA
so lets rewrite
sin(A - B) as sin(A + (-B) )
which now is
sinAcos(-B) + cosAsin(-B)
= sinAcosB - cosAsinB
DONE!
Do the same for cos(A - B)
for sin 2A, write it as sin(A + A) and use our first identity
sin 2A
= sin(A+A)
= sinAcosA + cosAsinA
= 2sinAcosA
for cos 2A, do the same thing
cos2A = cos(A+A)
= cosAcosA - sinAsinA
= cos^2 A - sin^2 A
but we also know cos^2 A = 1 - sin^2 A
and sin^2 A = 1 - cos^2 A
replace those in your cos2A result to get
cos^2 A - sin^2 A
= cos^2 A - (1-cos^2 A(
= 2cos^2 A - 1
do the same thing to get the third version of cos 2A
thank you so much !!
To derive the result for sin(alpha-beta) using the given identity, we can start by using the angle subtraction formula for sine:
sin(alpha-beta) = sin alpha * cos beta - cos alpha * sin beta
We can prove this identity by using the following steps:
Step 1: Start with the compound angle formula for sine:
sin(x + y) = sin x * cos y + cos x * sin y
Step 2: Substitute alpha for x and -beta for y:
sin(alpha + (-beta)) = sin alpha * cos(-beta) + cos alpha * sin(-beta)
Step 3: Use the properties of cosine and sine functions:
sin(alpha + (-beta)) = sin alpha * cos beta - cos alpha * sin beta
Step 4: Since sin(alpha + (-beta)) is equal to sin(alpha - beta), we have:
sin(alpha - beta) = sin alpha * cos beta - cos alpha * sin beta
This proves the given identity. Keep in mind that this is a derivation based on the angle subtraction formula for sine.