Anonymous
Anonymous asked in Education & ReferenceHomework Help · 8 years ago

PreCalc 12 Help, Identities?

The question is a proof: sin(A-B)cosB+cos(A-B)sinB = sinA.. If someone could write out the left side and show me how using the identities you can change it into sinA it would help a lot. Thanks :)

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  • Grid
    Lv 7
    8 years ago
    Favorite Answer

    This one is true. Your previous example was not, you might want to re-look that one.

    working on the LHS:

    sin(A-B)cosB + cos(A-B)sinB

    use following formulas:

    sin(A-B) = sin(A)cos(B) - cos(A)sin(B)

    cos(A - B) = sin(A)sin(B) + cos(A)cos(B)

    [sin(A)cos(B) - cos(A)sin(B)] * cos(B) + [sin(A)sin(B) + cos(A)cos(B)] * sin(B)

    sinAcos^2B - cosAsinBcosB + sinAsin^2B + cosAcosBsinB

    sinAcos^2B + sinAsin^2B

    sinA(sin^2B + cos^2B)

    sinA(1)

    sinA

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  • 8 years ago

    Recall:

    sin(x - y) = sin(x)cos(y) - cos(x)sin(y);

    cos(x - y) = cos(x)cos(y) + sin(x)sin(y);

    cos(x)^2 + sin(x)^2 = 1.

    sin(a - b)cos(b) + cos(a - b)sin(b)

    = (sin(a)cos(b) - cos(a)sin(b))cos(b) + (cos(a)cos(b) + sin(a)sin(b))sin(b)

    = sin(a)cos(b)^2 - cos(a)sin(b)cos(b) + cos(a)cos(b)sin(b) + sin(a)sin(b)^2

    = sin(a)cos(b)^2 + sin(a)sin(b)^2

    = sin(a)(cos(b)^2 + sin(b)^2)

    = sin(a).

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