and then there is .

could someone tell me how these sum and difference angle formulas are derived?

i've tried working it out but no luck. you don't have to do all of them, just one will suffice.

thanks

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- Mar 10th 2006, 07:36 PMc_323_hDerivation of Sum and Difference Angles

and then there is .

could someone tell me how these sum and difference angle formulas are derived?

i've tried working it out but no luck. you don't have to do all of them, just one will suffice.

thanks - Mar 11th 2006, 08:34 AMtopsquarkQuote:

Originally Posted by**c_323_h**

The picture:

On a typical xy coordinate system, sketch a line segment of unit length starting at the origin and at an angle . For convenience of sketching, place this segment in the first quadrant. Label the endpoint of this segment to be the point Q. . (Note: I don't know why this isn't LaTeX!)

Sketch another line segment of unit length in the second quandrant (again for convenience) starting from the origin and label its endpoint P. This will be at an angle measured from the +x axis. . (Note that so I don't need extra negative signs on the components.)

Finally, connect PQ with a line segment. This gives a triangle POQ (O is the origin). The angle POQ is .

The calculations:

We are going to get the length of PQ in two different ways. The first uses the above diagram in the given coordinate system. Label the coordinates of point P as and Q as and simply use the distance formula.

Finally we get:

.

New diagram:

More precisely, new coordinate system. Make a new set of x'y' axes, with the x' axis along the line segment OQ. (This merely has the effect of rotating the previous diagram so that OQ now lies along the x axis.) The coordinates of Q are now (1,0) and P are now . Again we find the length of PQ. (The line segments OQ, OP, and QP all have the same lengths as they did in the original coordinate system, of course.)

Which eventually comes to:

Equating our two values for PQ gives us:

** .

You can use this formula to be clever and find a number of relationships.

1. By setting in ** we get .

2. Letting in the expression in 1 gives .

3. Putting into ** we get .

4. Expanding the cosine term in 1 using ** and putting we get .

We can now use these to get the other sine and cosine relations:

Using ** to expand the cosine on the RHS and relations 3 and 4 give:

.

Starting with 1: and putting gives:

.

Now we use ** to expand the RHS:

.

And finally, we use 1 and 2 to get:

.

For the last formula:

We use the above expression for sine to expand this and 3 and 4 to get:

.

The tangent formulas come from using and dividing both the numerator and denominator by .

-Dan - Mar 11th 2006, 01:58 PMTD!
If you know how the trigonometric functions are defined by using complex exponentials, showing these formulas becomes easy.

We have

But because of the exponent laws and properties, we can rewrite

Working out and using iČ = -1 gives

Grouping real and imaginary parts

Then compare with what we started with

This yields

The formulas for the difference are easily derived by changing beta into -beta. The formula for tan follows from sin/cos. - Mar 11th 2006, 03:15 PMThePerfectHacker
Let me just state something useful. If you know any one of these formulas you can easily derive the other three know that sine and cosine are co-functions.

- Mar 12th 2006, 12:26 PMc_323_hQuote:

Originally Posted by**ThePerfectHacker**

- Mar 12th 2006, 07:20 PMc_323_hQuote:

Originally Posted by**ThePerfectHacker**