TRIGONOMETRIC FORMULAS

INTRODUCTION

Trigonometry is the study or measurement of triangles.   Central to this endeavor are the trigonometric formulas for the sine and cosine of the sum of two angles which can be used to derive all the other trigonometry formulas for some twenty in total.

SINE AND COSINE OF THE SUM OF TWO ANGLES

To start, we first draw two right triangles, and , each having one angle, α or β respectively, and then stack them one on the other as shown below.   We then draw the lines and to create the right triangle which also contains the angle α. Recalling the definition of the sides of a right triangle, as follows The vertical height of the triangle in the first figure is as follows But considering the triangles , , and we can also write  And so rearranging The horizontal width of triangle is given as And from the triangles , , and we can write  And so rearranging SINE AND COSINE OF THE DIFFERENCE OF TWO ANGLES

From simple geometric considerations we can write  These allow us to write the formulas for the difference between two angles as  TANGENT OF THE SUM AND DIFFERENCE OF TWO ANGLES

For the tangent, we can write so that we can write by dividing the top and bottom by , we get And likewise by noting we get the result DOUGLE ANGLE FORMULAS

If we set then we can write But from the Pythagorean Theorem we can also write so that we can write In like manner we can write the double angle sine formula as or And for the tangent we can write or by dividing top and bottom by we can write HALF ANGLE FORMULAS

If we set , then by recalling the relationship we can write  And from the relationship we can write TRIGONOMETRIC SUMS AS PRODUCTS

If we set and , then we can rewrite our original angles as  and substituting these into the formulas for the cosine of the sum and difference of two angles  And by adding and subtracting these two equations, we have  And using the formulas for the sine of the sum and difference of two angles  And by adding and subtracting these two equations, we have  TRIGONOMETRIC PRODUCTS AS SUMS

We can return to the original angles and rearrange the trigonometric sums as products on the left hand side.    FINAL NOTE

The discerning student will note an implicit assumption in that the sum of two angles is assumed to be less than 90 degrees.   While strictly correct, we can nevertheless extend the formulas for the sine and cosine of two angles, when that sum places the angle in the other three quadrants, without change.  From that point no further such assumptions need be made.