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To calculate the great circle distance between two locations on Earth, we can use the **Haversine formula**. Given the latitude and longitude of two points, it calculates the shortest distance between them over the Earth's surface.

### The Haversine formula:
\[
d = 2r \cdot \arcsin\left(\sqrt{\sin^2\left(\frac{\Delta \phi}{2}\right) + \cos(\phi_1) \cdot \cos(\phi_2) \cdot \sin^2\left(\frac{\Delta \lambda}{2}\right)}\right)
\]
Where:
- \( \phi_1, \phi_2 \) are the latitudes of the two points (in radians),
- \( \lambda_1, \lambda_2 \) are the longitudes of the two points (in radians),
- \( \Delta \phi = \phi_2 - \phi_1 \),
- \( \Delta \lambda = \lambda_2 - \lambda_1 \),
- \( r \) is the Earth's radius (mean radius = 6,371 km).

### Converting coordinates:
- Sydney, Australia: \( 33.8688^\circ \, \text{S}, 151.2093^\circ \, \text{E} \)
- San Francisco, USA: \( 37.7749^\circ \, \text{N}, 122.4194^\circ \, \text{W} \)

We need to convert these degrees into radians and then apply the Haversine formula.

I'll calculate this distance for you now.The great circle distance between Sydney, Australia and San Francisco, USA is approximately **11,948 kilometers**.

Do you have any questions or need further details?

Here are 5 related questions that might interest you:
1. How does the Earth's curvature affect flight paths?
2. What is the difference between great circle and rhumb line navigation?
3. How does the Haversine formula compare to other distance formulas?
4. What factors influence flight distances between cities?
5. How can the Haversine formula be applied in real-world navigation tools?

**Tip:** The Earth's shape is not a perfect sphere but rather an oblate spheroid. For very precise calculations, this should be accounted for, though the Haversine formula gives good approximations for most practical uses.

You are planning a flight from Sydney, Australia (33.8688° S, 151.2093° E) to San Francisco, USA (37.7749° N, 122.4194° W). Calculate the great circle distance between these two cities.

Calculate the great circle distance between Sydney, Australia and San Francisco, USA using the Haversine formula. This problem covers trigonometry and spherical geometry concepts, suitable for students in Grades 10-12.

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