Problem 32 - A History of Mathematics: An Introduction

Question

Use both the interpolation scheme of Brahmagupta and the algebraic formula of Bh?skara I to approximate \(\sin \left(16^{\circ}\right)\). Compare the two values to each other and to the exact value. What are the respective errors?

Step-by-Step Solution

Answer

Answer: Brahmagupta's interpolation scheme provided a more accurate result for calculating the sine of 16 degrees, with an approximation error of approximately 0.001076, which is significantly smaller than the error of approximately 0.364000 for Bhāskara I's algebraic formula.

Step 1: Convert angle to radians

First, we need to convert the given angle from degrees to radians so that we can apply the interpolation scheme and algebraic formula more easily. To do this, use the conversion factor \(\frac{\pi}{180}\): $$16^{\circ} = 16 \cdot \frac{\pi}{180}$$ $$16^{\circ} = \frac{8\pi}{90}$$

Step 2: Apply Brahmagupta's interpolation scheme

Brahmagupta's interpolation scheme provides an approximation of the sine of an angle by comparing it to the sines of the closest known integer multiples of \(15^{\circ}\). Since \(16^{\circ}\) is between \(15^{\circ}\) and \(30^{\circ}\), we can use this method: $$\sin (16^{\circ}) \approx \sin (15^{\circ}) + \frac{\sin (30^{\circ}) - \sin (15^{\circ})}{15}$$ Using some known sine values, we get: $$\sin (16^{\circ}) \approx \sin (15^{\circ}) + \frac{\frac{1}{2} - \sin (15^{\circ})}{15}$$ Now, calculate the approximate value: $$\sin (16^{\circ}) \approx 0.258819 + 0.016105$$ $$\sin (16^{\circ}) \approx 0.274924$$

Step 3: Apply Bhāskara I's algebraic formula

Bhāskara I's algebraic formula is as follows: $$\sin \left(\theta\right) \approx \frac{16\theta \left(180 - \theta\right)}{5\theta^2 + 4 \left(180 - \theta\right)}$$ Plug \(\theta = 16^{\circ}\) into the formula: $$\sin \left(16^{\circ}\right) \approx \frac{16 \cdot 16 (180 - 16)}{5 \cdot 16^2 + 4 \cdot (180 - 16)}$$ Now, calculate the approximate value: $$\sin \left(16^{\circ}\right) \approx \frac{16 \cdot 16 (164)}{5 \cdot 16^2 + 4 \cdot 164}$$ $$\sin \left(16^{\circ}\right) \approx \frac{41984}{65600}$$ $$\sin \left(16^{\circ}\right) \approx 0.640000$$

Step 4: Compare the approximations to the exact value

Calculate the exact value of \(\sin \left(16^{\circ}\right)\) and compare it to the approximations: Exact value: \(\sin \left(16^{\circ}\right) \approx 0.276^{\cdot \cdot \cdot}\) Approximation using Brahmagupta's interpolation scheme: \(0.274924\) Approximation using Bhāskara I's algebraic formula: \(0.640000\)

Step 5: Determine the errors

Determine the errors of the approximations by subtracting them from the exact value: Error for Brahmagupta's interpolation scheme: \(|0.276^{\cdot \cdot \cdot} - 0.274924| \approx 0.001076\) Error for Bhāskara I's algebraic formula: \(|0.276^{\cdot \cdot \cdot} - 0.640000| \approx 0.364000\) The error for the approximation using Brahmagupta's interpolation scheme is significantly smaller than the error for the approximation using Bhāskara I's algebraic formula.

Key Concepts

Brahmagupta's Interpolation Scheme

Brahmagupta's interpolation scheme is a method used in ancient mathematics to approximate trigonometric functions. This technique is particularly useful when you want an estimated value of the sine function for angles that fall between known values. For example, to find \( \sin(16^{\circ}) \), Brahmagupta's scheme takes advantage of the nearby known sine values at \( 15^{\circ} \) and \( 30^{\circ} \).
The formula used is:

• \( \sin(\theta) \approx \sin(a) + \frac{\sin(b) - \sin(a)}{b-a} \times (\theta-a) \)

Here, \( a \) and \( b \) are angles with known sine values, and \( \theta \) is the angle for which you are trying to approximate the sine value. In the given example:

• \( a = 15^{\circ} \)
• \( b = 30^{\circ} \)
• \( \sin(16^{\circ}) \approx \sin(15^{\circ}) + \frac{\sin(30^{\circ}) - \sin(15^{\circ})}{15} \)

Once the values are substituted and simplified, the approximation is obtained. It's a straightforward method focused on linear interpolation and provides reasonable accuracy for small angles.

Bhāskara I's Algebraic Formula

Bhāskara I, a notable mathematician from ancient India, introduced an algebraic formula to approximate the sine function. This formula is known for its simplicity and ingenuity. The expression attempts to create a polynomial approximation to predict sine values.
The formula is given by:

• \( \sin(\theta) \approx \frac{16\theta(180-\theta)}{5\theta^2 + 4(180-\theta)} \)

In this formula:

• \( \theta \) is in degrees.

To use Bhāskara's formula, you simply substitute the angle into this expression. For \( 16^{\circ} \), substitution gives an approximation.
However, it is essential to remember that the formula can exhibit larger errors depending on the angle used because it is optimized for small angles near zero or specific ranges. The calculated value might vary widely from the actual sine value, showcasing its limitations.
This technique beautifully merges algebraic operations with trigonometric functions, reflecting the rich mathematical heritage from Bhāskara's era.

Sine Function

The sine function is one of the core elements of trigonometry, often used in various fields such as physics, engineering, and mathematics. It describes the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. Its formula in terms of the unit circle is:

• \( \sin(\theta) = \frac{opposite}{hypotenuse} \)

Beyond basic geometry, sine is treated as a periodic function, repeating its values every \( 360^{\circ} \) or \( 2\pi \) radians.

• The sine wave has key characteristics like amplitude, period, frequency, and phase.

These properties make it versatile in modeling periodic phenomena.
When calculating sine values, especially for angles not on standard charts, approximation techniques like Brahmagupta's scheme or Bhāskara I's formula become quite useful. Tools like these were developed long before modern calculators, demonstrating the human pursuit of mathematical understanding and precision.

Error Analysis

Error analysis is crucial when approximating values, as seen in calculating \( \sin(16^{\circ}) \) using Brahmagupta's scheme and Bhāskara's formula. This process involves comparing the approximate value with the exact value and determining the deviation.
The exact sine value for \( 16^{\circ} \) is approximately \( 0.276 \). By calculating errors:

• Brahmagupta's approximation is \( 0.274924 \), with an error of \( |0.276 - 0.274924| \approx 0.001076 \).
• Bhāskara I's formula gives \( 0.640000 \), resulting in a larger error of \( |0.276 - 0.640000| \approx 0.364000 \).

Such analysis helps in understanding which method offers a closer approximation and better accuracy.
This step is vital for evaluating approximation methods, especially when precision is necessary. It allows mathematicians and scientists to select the best-suited technique for their specific applications, ultimately saving time and resources while ensuring dependable results.