Summation Of Series In Trigonometry

Sum of Series is an indispensable topic of mathematics. Series can be demonstrated as an expression in which the continuous or successive terms are written based on any formula and law. So, the sum of any successive terms is called the sum of series. In simple words, the addition of the terms written in the sequence is called the sum of the series. Apart from this, the sum of the series is subdivided into two parts. These are convergent series and divergent series. If the addition is done with the infinite series then it is convergent. Otherwise, it is a divergent series. 

Sum of Series in trigonometry 

The Sum of Series in trigonometry can be demonstrated as a sum of trigonometric functions by writing them in a sequence. In trigonometry, the sum of a series is considered a finite series. It is because the finite series contains the last term for addition. Similarly, in trigonometry, the sum up to the last term is performed to get accurate results. In general, there are two methods under trigonometry by which one can perform the Summation of trigonometric series. These two methods are as follows:

• Method of Differences

• C + iS Method.

The sum of any finite series in trigonometry remains finite because it contains the last terms. At the same time, the infinite series is endless. So, if one adds an infinite series number of times, even then, its result is hard to find. 

Method of Differences

In the procedure of Method of Differences, each trigonometric term is written as a difference between two trigonometric functions. It is written so that one trigonometric term contains the same function as the integer k. The other number contains trigonometric functions, either subtracted from r or added in r, i.e. (k+1) and (k-1).

Suppose, we need to find the sum of series 

v1 + v2 + v3 + … + vn,

Then for computing it with the Method of Differences, write it as a, 

By adding the 1 in integer 

vr = f(k+1) – f(k) 

By subtracting the 1 in integer 

vr = f(k) – f(k-1) 

By putting k = 1, 2, 3… so on in vr = f(k+1) – f(k).  

V1 = f(2) – f(1)

V2 = f(3) – f(2)

V3 = f(4) – f(3)….and so on. 

Vp-1 = f(p) – f(p-1)

Vp = f(p+1) – f(p)

So, on performing the addition, we get the series as, 

v1 + v2 + v3 +… + vn = f(p+1) – f(1)

C + iS Method

The C + iS method is also called a general method of solving the Sum of Series in trigonometry. Although, it is considered the easiest method of solving the sum of series in trigonometry. Using this method, one can find the sum of series with both finite and infinite functions. In this method of finding the sum of series, two trigonometric series are taken. 

Suppose we have taken the sine series and the cosine series. So, for adding the two series, firstly multiply the sine series with the I (iota). The iota work as a constant in the series. Although, an iota is an imaginary number whose value can be changed according to the demand of the question.  

Add the resulting expression with the cosine series after multiplying the iota in the sine series. 

The equation of the C + iS method is written as, 

C + iS

C = cosine series 

S = sine series

i = iota

Transformation of the last series 

The last series comes after computing the sum of the series with the Method of Differences and C + iS method. This last series is the resulting value of the computed series. One can convert these series into the other series. Trigonometric series are hard to read and understand so one can convert them into the other series for better understanding. Series in which the last series can be converted:

• Exponential Series

• Arithmetic Series

• Logarithmic Series

• Binomial series 

• Gregory Series

• Sine or Cosine Series

So, the last series obtained from the Method of Differences and C + iS method can convert into the above-written series. 

Conclusion

Sum of Series is widely used to perform addition in different series. Along with trigonometry, one can perform the addition of other series like Exponential Series, Arithmetic Series, Logarithmic Series, Binomial series and so forth. Although, one can convert the resulting series into another series. For instance, the resulting trigonometric series of the C + iS Method can be converted into the Arithmetic Series, Logarithmic Series, etc. There are two types of series, finite and infinite. The solution of finite series is well defined and easy to find because it contains the last term, but infinite series is not defined properly and is very hard to find.