Calculate crd(120◦), crd(150◦), crd(165◦), and crd(172 1
2
◦
)
using Hipparchus’s formula for crd(180◦ − α).

Hipparchus's formula for the chord of an angle \( \alpha \), or \( \text{crd}(\alpha) \), relates to the chord of the supplementary angle \( 180^\circ - \alpha \). The formula is:

\[
\text{crd}(180^\circ - \alpha) = 2R \sin\left( \frac{\alpha}{2} \right)
\]

where \( R \) is the radius of the circle. In the case of Hipparchus’s chord table, the radius \( R \) is generally normalized to \( 1 \), meaning that the formula simplifies to:

\[
\text{crd}(180^\circ - \alpha) = 2 \sin\left( \frac{\alpha}{2} \right)
\]

Now, we can calculate the chords for the specific angles using this formula.

### 1. For \( \text{crd}(120^\circ) \):

\[
\text{crd}(120^\circ) = 2 \sin\left( \frac{120^\circ}{2} \right) = 2 \sin(60^\circ)
\]
\[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
\]
\[
\text{crd}(120^\circ) = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3} \approx 1.732
\]

### 2. For \( \text{crd}(150^\circ) \):

\[
\text{crd}(150^\circ) = 2 \sin\left( \frac{150^\circ}{2} \right) = 2 \sin(75^\circ)
\]
Using \( \sin(75^\circ) \approx 0.9659 \):

\[
\text{crd}(150^\circ) = 2 \times 0.9659 = 1.9318
\]

### 3. For \( \text{crd}(165^\circ) \):

\[
\text{crd}(165^\circ) = 2 \sin\left( \frac{165^\circ}{2} \right) = 2 \sin(82.5^\circ)
\]
Using \( \sin(82.5^\circ) \approx 0.9914 \):

\[
\text{crd}(165^\circ) = 2 \times 0.9914 = 1.9828
\]

### 4. For \( \text{crd}(172.5^\circ) \):

\[
\text{crd}(172.5^\circ) = 2 \sin\left( \frac{172.5^\circ}{2} \right) = 2 \sin(86.25^\circ)
\]
Using \( \sin(86.25^\circ) \approx 0.9994 \):

\[
\text{crd}(172.5^\circ) = 2 \times 0.9994 = 1.9988
\]

### Summary of results:

- \( \text{crd}(120^\circ) \approx 1.732 \)
- \( \text{crd}(150^\circ) \approx 1.9318 \)
- \( \text{crd}(165^\circ) \approx 1.9828 \)
- \( \text{crd}(172.5^\circ) \approx 1.9988 \)

Would you like further clarification on these calculations or any other details?

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Here are 5 questions to deepen understanding:

1. How does Hipparchus’s chord formula relate to modern trigonometry functions?
2. What is the significance of calculating the chord in ancient astronomy?
3. How would you extend this method to compute other angles not listed here?
4. Can you explain why the value of \( \text{crd}(180^\circ - \alpha) \) approaches 2 as \( \alpha \) approaches 180°?
5. What differences are there between chord calculations and arc length measurements?

**Tip:** The chord of an angle is essentially a straight line connecting two points on the circle's circumference, and it has deep connections to modern sine functions.

Calculate crd(120°), crd(150°), crd(165°), and crd(172.5°) using Hipparchus’s formula for crd(180° − α).

Learn how to calculate chord lengths for angles 120°, 150°, 165°, and 172.5° using Hipparchus’s chord formula crd(180° − α) = 2 sin(α/2). This trigonometry problem covers concepts related to sine functions and chord calculations, suitable for Grades 10-12.

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