Trigonometry

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Prove the identity

sin(x+y+z)+sin(x+y-z)+sin(x-y+z)+ sin(x-y-z) = 4 sin(x)cos(y)cos(z)

This identity is so long and after i tried to expand the left side and it just looked something crap

Thanks for you help :)

  • Trigonometry -

    LS
    = sin((x+y)+z)+sin((x+y)-z)+sin((x-y)+z)+ sin((x-y)-z)
    = sin(x+y)cosz + cos(x+y)sinz
    + sin(x+y)cosz - cos(x+y)sinz
    + sin(x-y)cosz + cos(x-y)sinz
    + sin(x-y)cosz - cos(x-y)sinz

    = 2sin(x+y)cosz + 2sin(x-y)cosz
    = 2[cosz(sinxcosy + cosxsiny)] + 2[cosz(sinxcosy - cosxsiny)]

    = 2sinxcosycosz + 2sinycosxcosz + 2sinxcosycosz - 2 sinycosxcosz

    = 4 sinxcosycosz

    = RS

  • Trigonometry -

    I once had a great geometry teacher back in high school in Seattle. He was old and close to retirement. He used to say, to a mostly bored and unappreciative class: "Try to see the beauty in it." A few of us did. It was my favorite subject. Trig was next.

    Reiny's proof reminds me of that beauty.

  • Trigonometry -

    Thanks for the comment.

    Back in the days when we still wrote on a blackboards with chalk and we still had care-takers that would clean those boards at the end of the day....
    We once did a proof of "in any quadrilateral the largest area is obtained when opposite angles are supplementary" and it filled about 3 sections of blackboard.
    After it was done, the students drew a large picture frame around it.
    Next day it was still there with a note from the care-taker.
    "I looked at this, did not understand anything about it, but it sure looks like a piece of art"

    Your comment reminded me of that day, thanks.

  • Trigonometry -

    Well, i tried to see the beauty in trigonometry but to me, it is just too HARD!!!!!!!!

    There r so many formulae in trigonometry and how to i know which one to use.

    After i stay in the desk for 15 mins, i just wanna throw this stupid book away. I cant gain anything even i try so hard.

  • Trigonometry -

    A ladder that is 6 meters long is placed against a wall. It makes an angle 0f 34 degrees with the wall. Find how high up the wall it reaches and the distnce the base of the ladder is away from the wall

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