When working with trigonometric functions, one might come across a function with sin(x+y). However, upon viewing the outputs or the graph or this function, on realizes that you cannot simply distribute and that
sin(x+y) ≠ sin(x) + sin(y)
This is where Sum and difference formulas come in. We know that sin(x+y) isn't what is above. However, we don't know what it is yet, so we must derive it, or prove it without knowing what the other side of the equation might look like.
Deriving
Starting with sin(x+y), one can draw a picture as follows:
The picture depicts triangle ABC with angle x being added to triangle ACD with angle Y, thus providing side names to work sin(x+y) as
.
However, this is not a workable equation, as you cannot input anything into the sides as they are right now. We must rewrite until we have all x's and y's instead of sides.
1) Step one rewrites the equation as the sum of two lines as depicted in the picture, allowing for work with smaller triangles that share each of the sides.
2)Step two uses triangle FDC to replace side DF with DCcos(x).
3)Step three uses a property of rectangles to replace side FE with side CB.
4)Step four uses triangle ABC to replace side CB with ACsin(x).
5)Step five uses triangle ACD to replace DC with ADsin(y), and AC with ADcos(y), making all leading terms AD.
6)Step six divides AD from each term, leaving no side lengths and only x's and y's; a formula that can be worked.
Six Sum and Difference Formulas
Deriving each of the formulas this way could be a hassle. Therefor, since we have already come this far, each of the other five that exist can be derived directly from the first one. However, here are all six of them:
Application
These sum and difference formulas are really helpful in trigonometry. They help work through problems with angles smaller than those found on the unit circle. For example,
can be worked out as follows:
Thus giving what we set out to find, an exact measure for an angle that cannot be found on the unit circle.
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