Math Problem Statement

Use the Maclaurin series for 

cos(x)

 to compute 

cos(3°)

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

Solution

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

  1. Convert 33^\circ to radians because the Maclaurin series for trigonometric functions are typically in radians: 3=3π180=π603^\circ = \frac{3\pi}{180} = \frac{\pi}{60}

  2. The Maclaurin series for cos(x)\cos(x) is: cos(x)=n=0(1)nx2n(2n)!\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}

  3. Use the series to compute cos(π60)\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)\cos(x) are: cos(x)1x22!+x44!x66!+x88!\cos(x) \approx 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots

Let x=π60x = \frac{\pi}{60}:

Calculation

  1. x=π600.05235987755983x = \frac{\pi}{60} \approx 0.05235987755983

  2. First term: 11

  3. Second term: x22!=(0.05235987755983)220.0027415567780820.00137077838904-\frac{x^2}{2!} = -\frac{(0.05235987755983)^2}{2} \approx -\frac{0.00274155677808}{2} \approx -0.00137077838904

  4. Third term: x44!=(0.05235987755983)4240.00000750693412791240.00000031278808866\frac{x^4}{4!} = \frac{(0.05235987755983)^4}{24} \approx \frac{0.00000750693412791}{24} \approx 0.00000031278808866

  5. Fourth term: x66!=(0.05235987755983)67200.000000020562228997200.00000000002855864985-\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(π60)10.00137077838904+0.000000312788088660.00000000002855864985\cos\left(\frac{\pi}{60}\right) \approx 1 - 0.00137077838904 + 0.00000031278808866 - 0.00000000002855864985

cos(π60)10.00137077838904+0.00000031278808866\cos\left(\frac{\pi}{60}\right) \approx 1 - 0.00137077838904 + 0.00000031278808866

cos(π60)0.99862953439965\cos\left(\frac{\pi}{60}\right) \approx 0.99862953439965

Rounded to five decimal places: cos(3)0.99863\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)\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)\cos(x) to approximate cos(5)\cos(5^\circ)?
  4. What is the error bound for the Maclaurin series approximation of cos(x)\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)\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.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Maclaurin Series
Radians and Degrees Conversion

Formulas

Maclaurin series for cos(x)

Theorems

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Suitable Grade Level

Advanced High School and Above