Trigonometry/For Enthusiasts/Doing without Sine

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The IdeaEdit

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 formulasEdit

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 
 (2)  . 

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 consistentEdit

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":


Pythagorean identityEdit

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:




Sine FormulasEdit

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: