Calculate crd(120◦), crd(150◦), crd(165◦), and crd(172 21 using Hipparchus’s formula for crd(180◦ − α).

To calculate the chord lengths for the given angles using **Hipparchus's formula** for the chord of a supplementary angle \( \alpha \), we use the formula:

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

This formula connects the length of a chord subtended by an angle at the center of a circle with the sine function.

### Steps for each angle:

1. **crd(120°):**
   Using \( \alpha = 120^\circ \):

   \[
   \text{crd}(120^\circ) = 2 \times \sin \left( \frac{120^\circ}{2} \right) = 2 \times \sin(60^\circ)
   \]
   From trigonometric values, \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), so:

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

2. **crd(150°):**
   Using \( \alpha = 150^\circ \):

   \[
   \text{crd}(150^\circ) = 2 \times \sin \left( \frac{150^\circ}{2} \right) = 2 \times \sin(75^\circ)
   \]
   From trigonometric values, \( \sin(75^\circ) \approx 0.9659 \), so:

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

3. **crd(165°):**
   Using \( \alpha = 165^\circ \):

   \[
   \text{crd}(165^\circ) = 2 \times \sin \left( \frac{165^\circ}{2} \right) = 2 \times \sin(82.5^\circ)
   \]
   From trigonometric values, \( \sin(82.5^\circ) \approx 0.9914 \), so:

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

4. **crd(172.5°):**
   Using \( \alpha = 172.5^\circ \):

   \[
   \text{crd}(172.5^\circ) = 2 \times \sin \left( \frac{172.5^\circ}{2} \right) = 2 \times \sin(86.25^\circ)
   \]
   From trigonometric values, \( \sin(86.25^\circ) \approx 0.9981 \), so:

   \[
   \text{crd}(172.5^\circ) = 2 \times 0.9981 = 1.9962
   \]

### Summary of Results:
- \( \text{crd}(120^\circ) \approx 1.732 \)
- \( \text{crd}(150^\circ) \approx 1.9318 \)
- \( \text{crd}(165^\circ) \approx 1.9828 \)
- \( \text{crd}(172.5^\circ) \approx 1.9962 \)

Would you like further explanation or details on any of these calculations?

### Related Questions:
1. How does Hipparchus's formula relate to modern trigonometry?
2. What is the geometric interpretation of a chord in a circle?
3. How is the sine function used to calculate chord lengths?
4. Why does \( \sin(90^\circ) = 1 \) simplify calculations for large angles?
5. How would crd(90°) compare to crd(180° − α) for smaller angles?

### Tip:
When working with chords and trigonometry, remembering key sine values for specific angles (like 30°, 45°, 60°) simplifies calculations significantly.

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

Explore how to calculate the chord lengths for angles 120°, 150°, 165°, and 172.5° using Hipparchus’s formula. This step-by-step guide applies trigonometric concepts and formulas, specifically the sine function, to determine chord lengths. Suitable for students in Grades 10-12.

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