+142 In equilateral triangle ABC let points D and E trisect BC. Find sin DAE.
In equilateral triangle ABC let points D and E trisect BC. Find sin DAE.
It would be much easier if I knew how to draw on here, but I don't. So here goes. You should take a sheet of paper and draw as we go.
Draw the equilateral triangle. Label the vertexes. Mark your points D and E to trisect BC. Draw AD and AE.
Draw a perpendicular from angle A to the center of BC. Label the point on BC as P.
On my drawing, it looks like if you could get the sine of angle DAP and double it, you'd have the sine of DAE. I'm not certain of my footing about this assumption.
Assign a numerical value to each side AB, AC, and BC. It doesn't matter what the value is since we're going to be working with proportions, so why not call it 6 so at least some of the numbers will come out even.
You can figure the length of AP because it's the height of the original triangle.
You can figure the length of DP because it's half of DE which was a third of BC.
Use Pythagorean Theorem to figure the length of AD.
The sine of DAP is DP divided by AD.
Double the sine of DAP and you've got the sine of DAE. I think.
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+129849 Let the sides of the equilateral triangle be some convenient integer, say, 6
Draw altitude AF ......this makes DF = 1
Using the Law of Sines we have that
AF/ sin 60 = BA/ sin 90
AF / sin 60 = 6
AF = 6√3 / 2 = 3√3 = √27 =
And using the Pythagorean Theorem
DA = √[ AF^2 + DF^2] = √[ 27 + 1] = √28 = EA
And using the Law of Cosines
DE^2 = DA^2 + EA^2 - 2(DA)(EA) cos DAE
2^2 = 28 + 28 - (2*28) cos DAE simplify
4 - 56 = -56 cosDAE
-52 = -56 cosDAE
-52/-56 = cos DAE
13/14 = cosDAE
So....sin DAE = +√[ 1 - cos^DAE ] = + √[ 1 - (13/14)^2 ] = +√ [196 - 169] / 14 = √27 / 14 = (3/14)√3
