Use the Maclaurin series for 

cos(*x*)

 to compute 

cos(3°)

 correct to five decimal places. (Round your answer to five decimal places.)

To compute \(\cos(3^\circ)\) using the Maclaurin series and achieve accuracy to five decimal places, we can use the following steps:

1. Convert \(3^\circ\) to radians because the Maclaurin series for trigonometric functions are typically in radians:
   \[ 3^\circ = \frac{3\pi}{180} = \frac{\pi}{60} \]

2. The Maclaurin series for \(\cos(x)\) is:
   \[ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \]

3. Use the series to compute \(\cos\left(\frac{\pi}{60}\right)\) to the desired accuracy. 

We will compute several terms of the series until the added terms do not significantly change the result to five decimal places.

The first few terms of the Maclaurin series for \(\cos(x)\) are:
\[ \cos(x) \approx 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots \]

Let \( x = \frac{\pi}{60} \):

### Calculation
1. \( x = \frac{\pi}{60} \approx 0.05235987755983 \)

2. First term:
   \[ 1 \]

3. Second term:
   \[ -\frac{x^2}{2!} = -\frac{(0.05235987755983)^2}{2} \approx -\frac{0.00274155677808}{2} \approx -0.00137077838904 \]

4. Third term:
   \[ \frac{x^4}{4!} = \frac{(0.05235987755983)^4}{24} \approx \frac{0.00000750693412791}{24} \approx 0.00000031278808866 \]

5. Fourth term:
   \[ -\frac{x^6}{6!} = -\frac{(0.05235987755983)^6}{720} \approx -\frac{0.00000002056222899}{720} \approx -0.00000000002855864985 \]

Since the higher-order terms are increasingly small, we will add these terms together to find the approximation to five decimal places.

### Summing the terms:
\[ \cos\left(\frac{\pi}{60}\right) \approx 1 - 0.00137077838904 + 0.00000031278808866 - 0.00000000002855864985 \]

\[ \cos\left(\frac{\pi}{60}\right) \approx 1 - 0.00137077838904 + 0.00000031278808866 \]

\[ \cos\left(\frac{\pi}{60}\right) \approx 0.99862953439965 \]

Rounded to five decimal places:
\[ \cos(3^\circ) \approx 0.99863 \]

### Do you want details or have any questions?

### Further Questions:

1. What is the Maclaurin series for \(\sin(x)\)?
2. How does the accuracy of the Maclaurin series change with the number of terms included?
3. How can we use the Taylor series for \(\cos(x)\) to approximate \(\cos(5^\circ)\)?
4. What is the error bound for the Maclaurin series approximation of \(\cos(x)\)?
5. How do we convert between degrees and radians for trigonometric calculations?
6. How is the Maclaurin series derived for trigonometric functions?
7. Can we use the same method to find \(\cos(45^\circ)\) using the Maclaurin series?
8. What are the applications of using the Maclaurin series in physics and engineering?

### Tip:

When using series expansions to approximate values, always keep track of the error introduced by truncating the series, especially when high precision is required.

Learn how to compute cos(3°) accurately using the Maclaurin series approximation method. Step-by-step calculation with detailed terms and precision up to five decimal places. Suitable for advanced high school students and above.

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