Question

Verify that $$ \frac{\pi}{4}=4 \tan ^{-1}\left(\frac{1}{5}\right)-\tan ^{-1}\left(\frac{1}{239}\right) $$ a result discovered by John Machin in 1706 and used by him to calculate the first 100 decimal places of \(\pi .\)

Short answer

Using the arctangent addition and subtraction identities, the expression evaluates to \( \frac{\pi}{4} \).

Step-by-step solution

Understand the Arctangent Addition Formula

The arctangent addition formula states that \( \tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x + y}{1 - xy}\right) \) given \( xy < 1 \). This formula will be useful in verifying the expression given in the problem.

Apply the Formula to Each Term

Let's first express \( 4 \tan^{-1}\left(\frac{1}{5}\right) \) using the formula multiple times. We can split it into \( \tan^{-1}\left(\frac{1}{5}\right) + \tan^{-1}\left(\frac{1}{5}\right) + \tan^{-1}\left(\frac{1}{5}\right) + \tan^{-1}\left(\frac{1}{5}\right) \).

Use the Formula Repeatedly

Using the arctangent addition formula, compute the result of adding two \( \tan^{-1}\left(\frac{1}{5}\right) \) terms first. Repeatedly use the formula to find the result of four terms together.

Apply Arctangent Subtraction

Now that we have expressed \( 4 \tan^{-1}\left(\frac{1}{5}\right) \) as a single arctangent term, we use the arctangent subtraction formula: \( \tan^{-1}(a) - \tan^{-1}(b) = \tan^{-1}\left(\frac{a - b}{1 + ab}\right) \). Subtract \( \tan^{-1}\left(\frac{1}{239}\right)\) using this.

Simplify into Big Formula

Combine the results from the previous steps to express the given equation \( 4 \tan^{-1}\left(\frac{1}{5}\right) - \tan^{-1}\left(\frac{1}{239}\right) = \tan^{-1}(1)\). The \( \tan^{-1}(1) \) equals \( \frac{\pi}{4} \).

Key concepts

Trigonometric identities

Trigonometric identities are fundamental tools in mathematics that express relationships between the angles and sides of triangles through trigonometric functions like sine, cosine, and tangent. These identities help simplify complex expressions and solve various equations involving trigonometric functions. They are essential in calculus, physics, engineering, and many fields.In the context of the arctangent addition formula, we deal with the Inverse Tangent function, represented by \( \tan^{-1}(x) \). This identity, \( \tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x + y}{1 - xy}\right) \), is particularly useful when combining arctangent terms, provided the product \( xy < 1 \). This formula extends to multiple applications, making it easier to handle incremental sums of arctan angles. When coupled with its subtraction counterpart, it allows for precise calculations and simplifications in problems involving arctan expressions.

Arctangent subtraction

Arctangent subtraction is another powerful trigonometric tool that helps to manipulate inverse tangent expressions. It enables us to subtract one arctangent from another and is represented as: \( \tan^{-1}(a) - \tan^{-1}(b) = \tan^{-1}\left(\frac{a - b}{1 + ab}\right) \).Using this formula, if you already have an expression consisting of multiple arctangent terms, subtraction helps combine them efficiently. The condition \( ab < 1 \) ensures that the resulting expression remains valid. This subtraction is applied step-by-step in complex calculations like Machin's formula for \( \pi \) which involves the subtraction of \( \tan^{-1}\left(\frac{1}{239}\right)\) from \(4 \tan^{-1}\left(\frac{1}{5}\right)\). By calculating these combinations and differences accurately, one arrives at precise values required for advanced mathematical calculations.

Pi calculation methods

Calculating \( \pi \) has been a significant mathematical challenge through history, and several methods have been developed to achieve high precision. John Machin, in 1706, discovered a notable approach utilizing trigonometric properties, specifically arctangent identities, to calculate \( \pi \) to a high degree of accuracy.Machin's method involves the equation \( \frac{\pi}{4} = 4 \tan^{-1}\left(\frac{1}{5}\right) - \tan^{-1}\left(\frac{1}{239}\right)\). Breaking down this equation:

• The addition rule simplifies \(4 \tan^{-1}\left(\frac{1}{5}\right)\)
• The subtraction rule then combines it with \(- \tan^{-1}\left(\frac{1}{239}\right)\)
• This results in \( \tan^{-1}(1) \), equating to \( \frac{\pi}{4} \)

This method was revolutionary for its time because it allowed for highly precise computation of \( \pi \) using the relatively simple operations of addition and subtraction of arctangents, laying the groundwork for future mathematicians and computer-based calculations.