Database System Concepts
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ISBN: 9780078022159
Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher: McGraw-Hill Education
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![**Step 1: Taylor Series Expansions for Cosine and Sine**
Given the following Taylor series expansions for cosine and sine, create a program (using a for loop) to determine tan(x), where x is a value from 0 < x < π/2.
\[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots = \sum_{k=0}^{n} \frac{(-1)^k x^{2k}}{(2k)!} \]
\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots = \sum_{k=0}^{n} \frac{(-1)^k x^{2k+1}}{(2k+1)!} \]
This is a mathematical representation of the cosine and sine functions using their Taylor series. The Taylor series is an infinite sum of terms calculated from the values of the function's derivatives at a single point.
**Explanation of the Equations:**
1. **Cosine Series:**
- The cosine of x, denoted as cos(x), is approximated by an infinite sum.
- Each term in the series is derived from the derivatives of cos(x) at x = 0.
- The general term is \(\frac{(-1)^k x^{2k}}{(2k)!}\), where k is the term index.
2. **Sine Series:**
- The sine of x, denoted as sin(x), is also represented by an infinite sum.
- Similarly, each term is derived from the derivatives of sin(x) at x = 0.
- The general term is \(\frac{(-1)^k x^{2k+1}}{(2k+1)!}\), where k is the term index.
These series provide a way to compute trigonometric functions for values of x using polynomial approximations, which can be useful in writing programs to calculate these functions numerically.](https://content.bartleby.com/qna-images/question/1dfec151-558e-4c66-9085-1fd1afee3454/bb296b16-667c-4b4d-a574-06b2f0b8f7b3/nmfmgq_processed.png)
Transcribed Image Text:**Step 1: Taylor Series Expansions for Cosine and Sine**
Given the following Taylor series expansions for cosine and sine, create a program (using a for loop) to determine tan(x), where x is a value from 0 < x < π/2.
\[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots = \sum_{k=0}^{n} \frac{(-1)^k x^{2k}}{(2k)!} \]
\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots = \sum_{k=0}^{n} \frac{(-1)^k x^{2k+1}}{(2k+1)!} \]
This is a mathematical representation of the cosine and sine functions using their Taylor series. The Taylor series is an infinite sum of terms calculated from the values of the function's derivatives at a single point.
**Explanation of the Equations:**
1. **Cosine Series:**
- The cosine of x, denoted as cos(x), is approximated by an infinite sum.
- Each term in the series is derived from the derivatives of cos(x) at x = 0.
- The general term is \(\frac{(-1)^k x^{2k}}{(2k)!}\), where k is the term index.
2. **Sine Series:**
- The sine of x, denoted as sin(x), is also represented by an infinite sum.
- Similarly, each term is derived from the derivatives of sin(x) at x = 0.
- The general term is \(\frac{(-1)^k x^{2k+1}}{(2k+1)!}\), where k is the term index.
These series provide a way to compute trigonometric functions for values of x using polynomial approximations, which can be useful in writing programs to calculate these functions numerically.
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