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Subject: "(1)sine and (2)cos proofs of Pythagorean Theorem"     Previous Topic | Next Topic
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Conferences Educational Press Mathematics and general education Topic #11
Reading Topic #11
wongcheokwan
guest
Jan-19-07, 07:31 AM (EST)
 
"(1)sine and (2)cos proofs of Pythagorean Theorem"
 
   Thanks for Alexander Bogomolny's kind reply to my
previous note.

1)cosine proof:With the usual diagrams and notations.
In TRIANGLE ABC,angle C=90deg.
side a is opposite Angle A
side b is opp angle B
side c is opp angle C
From angle C drop a perpendicular to line AB at D.
BD=a.cos.B.. DA=c-a.cos.B
cos.A=b/c=(c-a.cos.B)/b....and cos.B=a/c
:: b.b=c.c-ac.cos.B =c.c-ac.a/c
:: b.b=c.c-a.a ::b2 +a2=c2

2)sine proof:BD=a.sin.A ...angle DCB is size A
angle CDB is 90.deg.
DA=BA-BD...Angle ACB is of sizeB.
sin.B=DA/AC=(c-a.sin.A)/b=b/c
sin.A=a/c ::cc-ca.a/c=bb ::cc-aa=bb

these steps are derived from similar triangles.

Thanks to readers


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