Calculate crd(30◦), crd(15◦), and crd(721) using the half-angle formula of Hipparchus, beginning with the fact that crd(60◦) = R = 60.



The **chord function** (denoted as \(\text{crd}(\theta)\)) was used by ancient Greek mathematicians like Hipparchus to describe the length of the chord subtended by a central angle \(\theta\) in a circle. For a circle of radius \(R = 60\) units (as used in ancient Greek astronomy), the chord function relates to modern trigonometry through:

\[
\text{crd}(\theta) = 2R \sin\left(\frac{\theta}{2}\right)
\]

Given that \(\text{crd}(60^\circ) = 60\), this confirms that \(R = 60\). To calculate \(\text{crd}(30^\circ)\), \(\text{crd}(15^\circ)\), and \(\text{crd}(7.21^\circ)\), we can use the **half-angle formula** for chords, which is based on the identity:

\[
\text{crd}(\theta) = 2R \sin\left(\frac{\theta}{2}\right)
\]

### 1. Calculate \(\text{crd}(30^\circ)\):
Using the half-angle formula with \(\text{crd}(60^\circ) = 60\) and \(R = 60\), the chord of \(30^\circ\) is:

\[
\text{crd}(30^\circ) = 2R \sin\left(\frac{30^\circ}{2}\right) = 2R \sin(15^\circ)
\]

But to calculate it step by step, we can apply a recursion:

\[
\text{crd}(30^\circ) = 60
\]



Calculate crd(30◦), crd(15◦), and crd(7.21◦) using the half-angle formula of Hipparchus, beginning with the fact that crd(60◦) = R = 60.

Learn how to calculate the chord lengths for angles 30°, 15°, and 7.21° using the half-angle formula of Hipparchus, starting from crd(60°) = 60. This guide walks you through the step-by-step application of the chord formula for a circle of radius R = 60, ideal for advanced trigonometry students.

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