Trigonometric R method
The trigonometric R method is a method of rewriting a weighted sum of sines and cosines as a single instance of sine (or cosine). This allows for easier analysis in many cases, as a single instance of a basic trigonometric function is often easier to work with than multiple are.
The R method is most often used to find the extrema (maximum and minimum) of combinations of trigonometric functions, since the extrema of a basic trigonometric function are easy to work with (both sine and cosine have a minimum of -1 and a maximum of 1).
Formal Statement and Proof
The R method can be used to convert an expression of the form into a single instance of either sine or cosine:
The sine form:
For any the following corresponds: where and orThe cosine form:
For any the following corresponds: where and or
First, let's prove the sine form. Let and Then the following equations correspond:
To prove the cosine form, let and Then the following equations correspond:
Using the R Method
Given the condition and what are the appropriate values of and for the following equation:
Since and we have
Therefore, and
Simplify using the R method.
(1) Using the sine form:
Since and we have(2) Using the cosine form:
Since and we have
Solve for :
Since and we have
Let Given we have In this interval the values of that satisfy are and Thus we have
Therefore the answer is
When using the R method, there are numerous possible values of and since sine and cosine are periodic functions. Adding to for any integer would give the same answer. Also, adding to for any integer and changing the sign of would also be equivalent to the answer. However we most often use and for convenience.
Applications to Extrema
The R method is frequently used to find the maximum or minimum of equations in the form of It is known that and for any real number so the maximum and minimum of are and respectively.
Find the sum of the maximum and minimum values of
Since we have This implies that
Therefore the answer is