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.