Ratios & Identities: Sum of sine/cos series where angles are in AP
Ratios & Identities: Sum of sine/cos series where angles are in AP
Trigonometric series involving the sum of sine or cosine functions where the angles are in arithmetic progression (AP) are common in mathematics, especially in the context of Fourier series, signal processing, and other applications. Understanding how to sum these series is crucial for solving many problems in trigonometry and calculus.
Arithmetic Progression (AP)
An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term. If the first term is ( a ) and the common difference is ( d ), then the ( n )-th term, ( a_n ), is given by:
[ a_n = a + (n-1)d ]
Sum of Sine Series where Angles are in AP
Consider a sine series where the angles are in AP:
[ S = \sin(a) + \sin(a+d) + \sin(a+2d) + \ldots + \sin(a+(n-1)d) ]
To find the sum of this series, we can use the trigonometric identity for the sum of two sines:
[ \sin(x) + \sin(y) = 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right) ]
By pairing terms from the beginning and the end of the series, we can apply this identity iteratively to find the sum.
Formula for Sum of Sine Series
The sum of the sine series can be expressed as:
[ S = \frac{\sin\left(\frac{nd}{2}\right)}{\sin\left(\frac{d}{2}\right)} \sin\left(a + \frac{(n-1)d}{2}\right) ]
Sum of Cosine Series where Angles are in AP
Similarly, consider a cosine series where the angles are in AP:
[ C = \cos(a) + \cos(a+d) + \cos(a+2d) + \ldots + \cos(a+(n-1)d) ]
We can use the trigonometric identity for the sum of two cosines:
[ \cos(x) + \cos(y) = 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right) ]
Formula for Sum of Cosine Series
The sum of the cosine series can be expressed as:
[ C = \frac{\sin\left(\frac{nd}{2}\right)}{\sin\left(\frac{d}{2}\right)} \cos\left(a + \frac{(n-1)d}{2}\right) ]
Table of Differences and Important Points
| Property | Sum of Sine Series | Sum of Cosine Series |
|---|---|---|
| Formula | ( S = \frac{\sin\left(\frac{nd}{2}\right)}{\sin\left(\frac{d}{2}\right)} \sin\left(a + \frac{(n-1)d}{2}\right) ) | ( C = \frac{\sin\left(\frac{nd}{2}\right)}{\sin\left(\frac{d}{2}\right)} \cos\left(a + \frac{(n-1)d}{2}\right) ) |
| First Term | ( \sin(a) ) | ( \cos(a) ) |
| Common Difference | ( d ) (for the angle) | ( d ) (for the angle) |
| Number of Terms | ( n ) | ( n ) |
| Pairing Method | Sine terms are paired from start and end | Cosine terms are paired from start and end |
Examples
Example 1: Sum of Sine Series
Find the sum of the series ( \sin(10^\circ) + \sin(20^\circ) + \sin(30^\circ) ).
Here, ( a = 10^\circ ), ( d = 10^\circ ), and ( n = 3 ).
Using the formula:
[ S = \frac{\sin\left(\frac{nd}{2}\right)}{\sin\left(\frac{d}{2}\right)} \sin\left(a + \frac{(n-1)d}{2}\right) ]
[ S = \frac{\sin\left(\frac{3 \times 10^\circ}{2}\right)}{\sin\left(\frac{10^\circ}{2}\right)} \sin\left(10^\circ + \frac{2 \times 10^\circ}{2}\right) ]
[ S = \frac{\sin(15^\circ)}{\sin(5^\circ)} \sin(20^\circ) ]
The exact value can be calculated using trigonometric tables or a calculator.
Example 2: Sum of Cosine Series
Find the sum of the series ( \cos(0^\circ) + \cos(30^\circ) + \cos(60^\circ) ).
Here, ( a = 0^\circ ), ( d = 30^\circ ), and ( n = 3 ).
Using the formula:
[ C = \frac{\sin\left(\frac{nd}{2}\right)}{\sin\left(\frac{d}{2}\right)} \cos\left(a + \frac{(n-1)d}{2}\right) ]
[ C = \frac{\sin\left(\frac{3 \times 30^\circ}{2}\right)}{\sin\left(\frac{30^\circ}{2}\right)} \cos\left(0^\circ + \frac{2 \times 30^\circ}{2}\right) ]
[ C = \frac{\sin(45^\circ)}{\sin(15^\circ)} \cos(30^\circ) ]
Again, the exact value can be calculated using trigonometric tables or a calculator.
Conclusion
The sum of sine and cosine series where the angles are in arithmetic progression can be calculated using specific formulas. These formulas are derived from trigonometric identities and the concept of pairing terms. Understanding these concepts is essential for solving problems in trigonometry and related fields.