Sinusoidal series

A sinusoidal series is defined as:

\[\begin{split}\begin{array}{l} x_0 = 1 \\ x_{i+1} = 2 \sin x_i+ 3 \cos x_i, i>=0 \end{array}\end{split}\]

This sequence begins with two positive terms and continues with negative terms. The first 10 values (rounded to 2 decimals) are: 1, 3.30, -3.28, -2.69, -3.57, -1.89, -2.83, -3.47 -2.21 -3.39.

Save both functions into file sinusoidal.py.

1. Write the function sinusoidal1() that given an integer \(k\), computes the sum of the elements of the previous series to the kth term (included), that is, \(x_0 +x_1 +...+x_k\). Examples:

   >>> round(sinusoidal1(9), 2)
   -19.02
   >>> round(sinusoidal1(10),2)
   -21.44
   >>> round(sinusoidal1(100),2)
   -280.47
   >>> round(sinusoidal1(345),2)
   -975.5
   >>> round(sinusoidal1(1000),2)
   -2837.07
   >>> round(sinusoidal1(2674),2)
   -7561.88
   >>> round(sinusoidal1(1000000),2)
   -2830726.31
   

   Note

   You can download the file with tests test-sinusoidal1.txt

2. Write the function sinusoidal2() that given a real value smax, tolerance epsilon, returns the position of the term in the previous series that is equal to the given value with tolerance epsilon. To determine the equality between two real numbers you must check if the absolute value of the difference between them is less than epsilon. Examples:

   >>> sinusoidal2(-3.5, 0.01)
   66
   >>> sinusoidal2(-3.5, 0.001)
   118
   >>> sinusoidal2(-2.9, 0.001)
   1767
   >>> sinusoidal2(-1.99, 0.01)
   23
   
   

   Note

   You can download the file with tests test-sinusoidal2.txt

Solution

A solution of these functions is provided in file sinusoidal.py