Will you please help solve this right triangle? (trig)?

Question

Suppose each edge of the cube is x cm long. Find the Sin and Cos of the diagonals formed by CF and CH.

Picture at http://pic90.picturetrail.com/VOL2297/9985690/18013441/282239262.jpg

use angle FCH

Answer 1

So you are going to make a right triangle with points C, H, and F where CF is the hypotenuse. All you know is that FH is x long because all the edges of the cube are x long.

To find the length of CH, you need to use pythagorean's theorem. CD and DH can be the sides of a right triangle where CH is the hypotenuse. You know CD = DH = x because all the edges are x long. So x² + x² = CH².

That means 2x² = CH² .. so .. CH = x(sqrt (2)) ... read that last part as x times the square root of 2.

Now we have the two sides of the triangle I mentioned at the beginning and we need the length of the hypotenuse. Again, time to use pythagorean's theorem.

FH = x ... CH = x(sqrt(2)) ... and CF = ???. We know that CF is the hypotenuse, so here we go again!

FH² + CH² = CF² ... x² + (x(sqrt (2))² = CF² ...
x² + 2x² = CF² ... 3x² = CF² ...
so .... x(sqrt(3)) = CF.

Shew .. now we have the lengths of the of all the sides of the triangle. The problem here is that you can't find sin or cos of anything without knowing which ANGLE you are referencing. So without some more information, I am not going to be able to continue with the question. Hopefully this explanation helps.

Answer 2

CH ² = 2 x ²
CH = √2 x
CF ² = 2 x ² + x ² = 3 x ²
CF = √3 x
Sin C = x / √2 x = 1 / √2
Cos C = √2 x / √3 x = √2 / √3