Question

Have you ever wondered how a calculator or computer program evaluates the sine, cosine, or tangent of a given angle? The calculator or computer program approximates these values using a power series. The terms of a power series contain ascending positive integral powers of a variable. The more terms in the series, the more accurate the approximation. With a calculator in radian mode, verify the following for small values of \(x,\) for example, \(x=0.5\). a) \(\tan x=x+\frac{x^{3}}{3}+\frac{2 x^{5}}{15}+\frac{17 x^{7}}{315}\) b) \(\sin x=x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}\) c) \(\cos x=1-\frac{x^{2}}{2}+\frac{x^{4}}{24}-\frac{x^{6}}{720}\)

Short answer

For all functions, the computed values using the power series closely match the calculator values: \(\tan(0.5)\approx 0.5463, \(\sin(0.5)\approx 0.4794, \(\cos(0.5)\approx 0.8776.

Step-by-step solution

Calculate \(\tan(0.5)\) Using Calculator

First, make sure your calculator is in radian mode. Then input \(\tan(0.5)\), which should give you approximately \0.5463\.

Compute \(\tan(0.5)\) Using the Power Series

Using the series \tan x = x + \frac{x^{3}}{3} + \frac{2 x^{5}}{15} + \frac{17 x^{7}}{315}\, substitute \(x=0.5\). Calculate each term: \0.5 + \frac{(0.5)^3}{3} + \frac{2(0.5)^5}{15} + \frac{17(0.5)^7}{315}\ which results in approximately \0.5463\.

Calculate \(\tan(0.5)\) Difference

Compare the values calculated in Steps 1 and 2. The difference should be very small, verifying the series approximation.

Calculate \(\tan(0.5)\) Using Calculator

Calculate \(\sin(0.5)\) with your calculator, ensuring it is in radian mode. The value should be approximately \0.4794\.

Compute \(\tan(0.5)\) Using the Power Series

Using the series \sin x = x - \frac{x^{3}}{6} + \frac{x^{5}}{120} - \frac{x^{7}}{5040}\, substitute \(x=0.5\). Calculate each term: \0.5 - \frac{(0.5)^3}{6} + \frac{(0.5)^5}{120} - \frac{(0.5)^7}{5040}\ which results in approximately \0.4794\.

Calculate \(\tan(0.5)\) Difference

Compare the values calculated in Steps 4 and 5. The difference should be very small, verifying the series approximation.

Calculate \(\tan(0.5)\) Using Calculator

Calculate \(\cos(0.5)\) with your calculator, ensuring it is in radian mode. The value should be approximately \0.8776\.

Compute \(\tan(0.5)\) Using the Power Series

Using the series \cos x = 1 - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720}\, substitute \(x=0.5\). Calculate each term: \1 - \frac{(0.5)^2}{2} + \frac{(0.5)^4}{24} - \frac{(0.5)^6}{720}\ which results in approximately \ 0.8776\.

Calculate \(\tan(0.5)\) Difference

Compare the values calculated in Steps 7 and 8. The difference should be very small, verifying the series approximation.

Key concepts

Power Series

A power series is a series of terms involving ascending positive integral powers of a variable. It can be written in the form: \[ \sum_{n=0}^{\infty} a_n x^n \] where \a_n\ are coefficients and \x\ is the variable. Power series are used in approximating functions including trigonometric functions. The more terms you include from a power series, the closer the approximation is to the actual function value. This concept is crucial for understanding how calculators quickly compute values of functions like sine, cosine, and tangent for any angle you input.

Sine Function Approximation

The sine function can be approximated using its power series expansion. For small angles, let's take the series up to the fourth term. The series is: \[ \sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} \]
For example, if we want to approximate \sin(0.5)\:
\[ \sin(0.5) \approx 0.5 - \frac{(0.5)^3}{6} + \frac{(0.5)^5}{120} - \frac{(0.5)^7}{5040} \]
This results in approximately \0.4794\. Notice how close this approximation is to the true value you would get from a calculator.

Cosine Function Approximation

The cosine function can also be approximated using its power series. The series for cosine is: \[ \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} \]
To approximate \cos(0.5)\, we substitute \x=0.5\ into the series:
\[ \cos(0.5) \approx 1 - \frac{(0.5)^2}{2} + \frac{(0.5)^4}{24} - \frac{(0.5)^6}{720} \]
This results in approximately \0.8776\. Just like with sine, this approximation is very close to what you would find using a calculator.

Tangent Function Approximation

The tangent function has its own power series, which can be used for approximation. The series is: \[ \tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} \]
For example, to approximate \tan(0.5)\:
\[ \tan(0.5) \approx 0.5 + \frac{(0.5)^3}{3} + \frac{2(0.5)^5}{15} + \frac{17(0.5)^7}{315} \]
This results in approximately \0.5463\. By comparing this result with the one obtained using a calculator, you can see that the series provides a very accurate approximation.