We know that:
So do we really need the function?
Or put another way, could we have worked out all our interesting formulas for things like in terms just of and then derived every formula that has a in it from that?
The answer is yes.
We don't need to have one geometric argument for and then do another geometric argument for . We could get our formulas for directly from formulas for
Angle Addition and Subtraction formulas
To find a formula for in terms of and : construct two different right angle triangles each drawn with side having the same length of one, but with , and therefore angle . Scale up triangle two so that side is the same length as side . Place the triangles so that side is coincidental with side , and the angles and are juxtaposed to form angle at the origin. The circumference of the circle within which triangle two is embedded (circle 2) crosses side at point , allowing a third right angle to be drawn from angle to point . Now reset the scale of the entire figure so that side is considered to be of length 1. Side coincidental with side will then be of length , and so side will be of length in which length lies point . Draw a line parallel to line through the right angle of triangle two to produce a fourth right angle triangle, this one embedded in triangle two. Triangle 4 is a scaled copy of triangle 1, because:
(1) it is right angled, and
The length of side is as . Thus point is located at length:
giving us the "Cosine Angle Sum Formula".
Proof that angle sum formula and double angle formula are consistent
We can apply this formula immediately to sum two equal angles:
From the theorem of Pythagoras we know that:
in this case:
Substituting into (I) gives:
which is identical to the "Cosine Double Angle Sum Formula":
Armed with this definition of the function, we can restate the Theorem of Pythagoras for a right angled triangle with side c of length one, from:
We can also restate the "Cosine Angle Sum Formula" from:
The price we have to pay for the notational convenience of this new function is that we now have to answer questions like: Is there a "Sine Angle Sum Formula". Such questions can always be answered by taking the form and selectively replacing by and then using algebra to simplify the resulting equation. Applying this technique to the "Cosine Angle Sum Formula" produces:
-- Pythagoras on left, multiply out right hand side
-- Carefully selected Pythagoras again on the left hand side
-- Multiplied out
-- Carefully selected Pythagoras
-- Algebraic simplification
taking the square root of both sides produces the "Sine Angle Sum Formula"
We can use a similar technique to find the "Sine Half Angle Formula" from the "Cosine Half Angle Formula":
We know that , so squaring both sides of the "Cosine Half Angle Formula" and subtracting from one:
So far so good, but we still have a to get rid of. Use Pythagoras again to get the "Sine Half Angle Formula":
or perhaps a little more legibly as: