In 2 dimensions, when you look at an angle it is obvious what size it is. This is more difficult in 3 dimensions. Here's an experiment you can try that will show what I mean:

Cut a triangle ABC out of a sheet of paper. If you can, make it fairly large and make angle A obtuse. Of course, B and C will be acute.

Now go to the corner of your room and put A into the corner. Juggle with the position of B and C until AB lies against one wall and AC lies against the other. (hint: Out of B and C, one will need to be lower down the wall than A and the other higher.)

Now try to fit B into the corner. (This time both A and C will need to be lower than B, with BA touching one wall and BC touching the other.)

This will work whatever the size of the angles. So what is the true angle between the walls? By choosing A at the corner, B in one wall and C in the other, I have shown that any obtuse angle will 'fit'. With B at the corner, I have shown that any acute angle will fit.

But, you know that the angle between the walls is actually 90. So how can we 'see' this true angle and measure the angle between faces correctly.

Look at my diagram below. You will 'see' the angle between the walls as 90 if you are looking along the dotted line. When you look in the direction of the arrow, the angle looks like it is 90 as it should.

To do the same for the angle between faces on a tetrahedron you need to imagine you are looking along the line of intersection between two faces.

In that other thread for faces VAB and VBC, the line to look along is VB. The true angle between the faces is the size of angle AGC, where G is on VB and both AG and CG are at right angles to VB.

The second diagram in that thread (post 3) shows the true shape of face VBC. So my method is to work out CG from that. For a regular tetrahedron the angle B is 60. AG will be the same length as CG.

Then you can use the cosine rule to calculate AGC.

Bob

]]>use tan = sin/cos and sec = 1/cos

Re-write the equation in the form R(cosBcosA - sinBsinA) where B can be determined.

Hence get cos(A+B)

EDIT: better to try this

Write as sinA = function of cosA

Square and form a quadratic in cos using sin^2 = 1 - cos^2

Bob

]]>If A is an acute angle such that \tan A + \sec A = 2, then find \cos A.

]]>What is the cosine of the angle between two adjacent faces of a regular tetrahedron?

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