Question

Calculate crd \(\left(30^{\circ}\right), \operatorname{crd}\left(15^{\circ}\right)\), and \(\operatorname{crd}\left(7 \frac{1}{2}^{\circ}\right)\) using the halfangle tormula of Hipparchus, beginning with the fact that \(\operatorname{cod}\left(60^{2}\right)=R=60 .\)

Short answer

Question: Calculate the chord lengths for angles 30, 15, and 7.5 degrees using the half-angle formula of Hipparchus if the cod function value for 60 degrees squared is equal to 60. Answer: Using the half-angle formula of Hipparchus, the chord lengths are approximately as follows: - crd(30°) ≈ 31.1769 - crd(15°) ≈ 15.58865 - crd(7.5°) ≈ 7.794379

Step-by-step solution

Half-angle formula and chord function relation

Recall the half-angle formula for the chord function (Hipparchus formula): \(crd\left(\frac{\alpha}{2}\right) = \sqrt{(1-\operatorname{cod}(\alpha))(1+\operatorname{cod}(\alpha))}\) and the relationship between cod and crd: \(\operatorname{cod}(\alpha) = 1-\frac{\operatorname{crd}^{2}(\alpha)}{2R^{2}}\)

Calculate crd(30) using given cod(60) value and half-angle formula

We know \(\operatorname{cod}(60^{2}) = R = 60\). This means: \(\operatorname{cod}(60) = \sqrt{60}\). Now apply the half-angle formula: \(crd\left(30\right) = \sqrt{(1-\sqrt{60})(1+\sqrt{60})}\) Calculate the value of crd(30): \(crd\left(30\right) \approx 31.1769\)

Calculate crd(15) using crd(30) and half-angle formula

To find crd(15), we first need to get cod(30). Use the relationship between cod and crd: \(\operatorname{cod}(30) = 1-\frac{\operatorname{crd}^2(30)}{2R^2} = 1-\frac{31.1769^2}{2(60)^2} \approx 0.154215\) Now apply the half-angle formula: \(crd\left(15\right) = \sqrt{(1-0.154215)(1+0.154215)}\) Calculate the value of crd(15): \(crd\left(15\right) \approx 15.58865\)

Calculate crd(7.5) using crd(15) and half-angle formula

To find crd(7.5), we first need to get cod(15). Use the relationship between cod and crd: \(\operatorname{cod}(15) = 1-\frac{\operatorname{crd}\left(15\right)^2}{2R^2} = 1-\frac{15.58865^2}{2(60)^2} \approx 0.573606\) Now apply the half-angle formula: \(crd\left(7.5\right) = \sqrt{(1-0.573606)(1+0.573606)}\) Calculate the value of crd(7.5): \(crd\left(7.5\right) \approx 7.794379\)

Conclude with the calculated chord lengths

Using the half-angle formula of Hipparchus, we have calculated the following chord lengths: - crd\(\left(30^{\circ}\right) \approx 31.1769\) - crd\(\left(15^{\circ}\right) \approx 15.58865\) - crd\(\left(7.5^{\circ}\right) \approx 7.794379\)

Key concepts

Hipparchus Formula

The Hipparchus formula plays a pivotal role in understanding the relationship between the lengths of chords in a circle and the angles they subtend at the center. Named after the ancient Greek astronomer and mathematician Hipparchus, this formula is a precalculus method used for measuring astronomical distances and is foundational in trigonometry.

To elucidate, consider a circle of radius R, and let's examine a chord that subtends an angle \( \alpha \) at the circle's center. The formula states that the length of the chord, denoted as \( \operatorname{crd}(\alpha) \), can be expressed in terms of the angle using the formula \( \operatorname{crd}(\alpha) = 2R \sin(\alpha/2) \).

This relation is especially useful when using the half-angle formula to find the length of a chord given a particular angle. A student attempting to follow these calculations should understand the basic sine function and how it connects an angle with the length of a chord. This knowledge is critical when using Hipparchus’ half-angle formula to divide angles and successively find chord lengths, as illustrated in the textbook exercise.

Half-Angle Formula

The half-angle formula is an essential tool in trigonometry, greatly simplifying the process of calculating the chords for smaller angles based on those of larger angles. In essence, this formula allows us to find the sine of half an angle if we know the sine of the original angle. The formula is given mathematically as \( \sin(\frac{\alpha}{2}) = \pm\sqrt{\frac{1-\cos(\alpha)}{2}} \).

To connect this with the chord function from our problem, one can derive the chord's length for half the angle using the Hipparchus formula mentioned earlier: \( crd\left(\frac{\alpha}{2}\right) = \sqrt{(1-\operatorname{cod}(\alpha))(1+\operatorname{cod}(\alpha))} \). This method was employed in the exercise to calculate the chord lengths of \( 15^\circ \) and \( 7.5^\circ \) based on knowing the length for \( 30^\circ \).

Understanding the half-angle formula is crucial for anyone studying mathematics, especially those interested in fields that involve computation of trigonometric values, such as physics, engineering, or even computer graphics.

History of Mathematics

Tracing the history of mathematics reveals a rich tapestry of intellectual advancement and cultural exchange, showcasing the importance of seminal figures like Hipparchus. The development of mathematical concepts such as the half-angle formula and chord lengths is deeply rooted in the needs of ancient civilizations to solve practical problems like navigation, construction, and astronomy.

Hipparchus, who lived around the 2nd century BCE, made significant contributions to trigonometry and is often considered the father of this branch of mathematics. His work on chords, which was a precursor to the modern sine function, laid the groundwork for future mathematicians to build upon. As learners engage with mathematical exercises that involve Hipparchus' formula or the half-angle formula, they're not just solving problems; they're participating in a tradition that dates back millennia and spans many cultures from the Babylonians and Egyptians to the Greeks and Indians.

Understanding the historical context can help students appreciate the depth and breadth of mathematics as a human endeavor and inspire them to be curious about how mathematical principles they study today came to be and how they have been used throughout history.