Trigonometric Formulas

 

 

INTRODUCTION

 

Trigonometry is the study of triangles and especially right triangles.   Central to this endeavor are the sine and cosine of the sum of two angles.  These two formulas can then 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,  with an angle of .  We then draw the lines  and   to create the right triangle , which also contains the angle .

 

 

Please recall that if we know the hypotenuse “r” and one other angle “x” of a right triangle, then we can calculate lengths of the other two sides as follows

Considering the triangle  we can write

 

   

 

And finally rearranging

 

 

Again considering the triangle , 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 using the formula above we can write

 

 

From the Pythagorean Theorem for the above triangle with r=1, we can write

 

 

Combining these two gives us

 

 

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, i.e.  , is less than 90 degrees.   While strictly correct, we can nevertheless use the formulas for the sine and cosine for angles greater and than 90 degrees for all the various combinations.  The final result is exactly the same. From that point no further such assumptions need be made.