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.