Posted by Tommy on .
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 :)
= 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
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.
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.
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.
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