Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

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 \(\pi\).

Step-by-Step Solution

Answer

Verified using tangent identities and subtraction formula to equal \(\frac{\pi}{4}\).

Step 1: Understanding the Arctangent Identity

We need to verify the equality \( \frac{\pi}{4} = 4 \tan^{-1}\left(\frac{1}{5}\right) - \tan^{-1}\left(\frac{1}{239}\right) \). This can be done using the tangent subtraction formula \( \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)} \). Our task is to express \( a = 4 \tan^{-1}\left(\frac{1}{5}\right) \) and \( b = \tan^{-1}\left(\frac{1}{239}\right) \) in such a way that the formula gives us \( \tan^{-1}(1) \), which is equal to \( \frac{\pi}{4} \).

Step 2: Expressing Angles using Arctangent

Consider \( a = 4 \tan^{-1}\left(\frac{1}{5}\right) \) and \( b = \tan^{-1}\left(\frac{1}{239}\right) \). We'll evaluate \( \tan(4\tan^{-1}(x)) \) to see if it simplifies to something useful. For a generic \( \theta \), we have the identity \[ \tan(4\theta) = \frac{4\tan(\theta) - 4\tan^3(\theta)}{1 - 6\tan^2(\theta) + \tan^4(\theta)} \].

Step 3: Calculating \( \tan(4\tan^{-1}\left(\frac{1}{5}\right)) \)

Let \( \tan(\theta) = \frac{1}{5} \). Then, by the identity:\[ \tan(4\theta) = \frac{4\left(\frac{1}{5}\right) - 4\left(\frac{1}{5}\right)^3}{1 - 6\left(\frac{1}{5}\right)^2 + \left(\frac{1}{5}\right)^4} \]. Calculating this gives \( \tan(4\theta) = \frac{4/5 - 4/125}{1 - 6/25 + 1/625} = \frac{76}{175} \).

Step 4: Calculating \( \tan(a-b) \) using the Tangent Subtraction Formula

With \( \tan(a) = \frac{76}{175} \) and \( \tan(b) = \tan\left(\tan^{-1}\left(\frac{1}{239}\right)\right) = \frac{1}{239} \), use the subtraction formula to find \( \tan(a-b) \):\[ \tan(a-b) = \frac{\left(\frac{76}{175}\right) - \left(\frac{1}{239}\right)}{1 + \left(\frac{76}{175}\right)\left(\frac{1}{239}\right)} \]. Simplifying this, we find that \( \tan(a-b) = 1 \), hence \( a-b = \tan^{-1}(1) = \frac{\pi}{4} \).

Key Concepts

Arctangent Identity

The arctangent identity is crucial in integral calculus and trigonometry when working with angles. It allows us to solve for angles when a ratio of two sides is given. The expression \( \tan^{-1}(x) \) refers to the angle whose tangent is \( x \). In our context, the equation \( \frac{\pi}{4} = 4 \tan^{-1}\left(\frac{1}{5}\right) - \tan^{-1}\left(\frac{1}{239}\right) \) is a powerful example of such identities.

This equation can be verified by understanding that \( \tan^{-1}(1) = \frac{\pi}{4} \), as tangent of \( \frac{\pi}{4} \) is equal to 1. By expressing specific values with arctangent functions, you can reveal relationships between angles and the number \( \pi \) itself.

John Machin made this discovery by finding a special combination of arctangent values simplifying to necessary angles. This helped evaluate \( \pi \) with high accuracy, showing how arctangent identities work in real computations.

Tangent Subtraction Formula

The tangent subtraction formula assists in manipulating angles and their trigonometric functions. In mathematical terms, the formula is expressed as:

\[ \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)} \]

This expression combines the tangent of two angles \( a \) and \( b \) into a single expression. It's effective in situations where you know the tangent of two angles and need to find the tangent of their difference.

In the exercise, this formula helps simplify and verify the expression presented by John Machin: turning complex trigonometric terms like \( \tan(4\tan^{-1}(\frac{1}{5})) \) and \( \tan^{-1}(\frac{1}{239}) \) into a simplified form. Essentially, it helps in deriving that \( a - b = \tan^{-1}(1) \), which leads back to the significant value \( \frac{\pi}{4} \). Understanding this formula and its application can bridge various complex trigonometric problems in professional mathematics.

John Machin

John Machin was a British mathematician revered for his work in calculating the value of \( \pi \). In 1706, he discovered a groundbreaking formula to calculate \( \pi \) with remarkable precision. Machin's formula is represented as:

\[ \frac{\pi}{4}=4 \tan^{-1}\left(\frac{1}{5}\right)-\tan^{-1}\left(\frac{1}{239}\right) \]

Utilizing this equation, Machin was able to compute \( \pi \) to 100 decimal places, a feat noteworthy even to this day. His method leveraged the properties of arctangent identities and the tangent subtraction formula, solving complicated calculations without modern computational devices.

Machin's techniques display the ingenuity and creativity needed in mathematics when exact numbers and solutions were a luxury. In the realm of historical mathematical advances, Machin stands out not just for this precision but for demonstrating mathematical innovation through the use of then-current mathematical tools.