Distance calculation - Free Printable

Let’s solve this step by step.

We are given two points on a sphere (like Earth):

- Point P₁ at latitude lat₁ and longitude lon₁
- Point P₂ at latitude lat₂ and longitude lon₂

We want to find the great-circle distance between them — that is, the shortest distance along the surface of the sphere.

The formula for great-circle distance uses the spherical law of cosines:

> cos(q) = sin(lat₁)·sin(lat₂) + cos(lat₁)·cos(lat₂)·cos(Δlon)

Where:
- q is the central angle between the two points (in radians)
- Δlon = |lon₁ - lon₂| (difference in longitudes)
- All angles must be in radians, not degrees!

Then, once we have q (in radians), the actual distance d is:

> d = R · q

Where R is the radius of the sphere (for Earth, usually about 6371 km).

But wait — looking at the diagram, it shows angles like “90° - lat” which suggests they’re using co-latitudes (angle from North Pole). That’s fine — but let’s stick with standard latitude/longitude formulas unless told otherwise.

Actually, the diagram labels the triangle sides as “90° - lat₁”, “90° - lat₂”, and the included angle as “Δlon”. So this is a spherical triangle with vertices at:

- North Pole
- P₁
- P₂

And side opposite the pole is the arc between P₁ and P₂ — which is exactly the great-circle distance q (in angular measure).

So yes — we can use the spherical law of cosines directly on this triangle:

In any spherical triangle with sides a, b, c (measured in angles) and angle C between sides a and b:

> cos(c) = cos(a)·cos(b) + sin(a)·sin(b)·cos(C)

In our case:

- Side a = 90° - lat₁ → co-latitude of P₁
- Side b = 90° - lat₂ → co-latitude of P₂
- Angle C = Δlon = |lon₁ - lon₂|
- Side c = q → the angular distance between P₁ and P₂

So plug into formula:

> cos(q) = cos(90° - lat₁)·cos(90° - lat₂) + sin(90° - lat₁)·sin(90° - lat₂)·cos(Δlon)

But remember trig identities:

- cos(90° - x) = sin(x)
- sin(90° - x) = cos(x)

So substitute:

> cos(q) = sin(lat₁)·sin(lat₂) + cos(lat₁)·cos(lat₂)·cos(Δlon)

Same as before! Good.

Now — here’s the catch: the problem doesn’t give us numbers. It just shows a diagram with variables.

Wait — maybe I missed something? Let me check again.

Looking back at the user input: there’s no numerical data provided. The image has labels like P₁(lat₁, lon₁), P₂(lat₂, lon₂), and angles labeled symbolically.

This means — perhaps the task is to derive or state the formula for great-circle distance based on the diagram?

But the instruction says: “Solve the problem accurately.” And “work through all calculations carefully.”

Hmm. Maybe the original problem had specific values, but they weren't included in the text? Or perhaps this is a conceptual question?

Wait — re-reading the user message: “You are an educational assistant helping a student solve homework problems. The user uploaded an image containing a task.”

But since I can’t see the image beyond what’s described, and the description only gives symbolic labels...

Perhaps the intended task is to write down the formula for great-circle distance using the diagram?

Alternatively — maybe in the actual image, there were numbers? But since none are provided here, I’ll assume the goal is to derive the formula shown in the diagram.

Let me think differently.

The diagram shows a spherical triangle with:

- Two sides: 90° - lat₁ and 90° - lat₂
- Included angle: Δlon = lon₂ - lon₁ (assuming lon₂ > lon₁)
- Opposite side: q (angular distance)

Using spherical law of cosines:

cos(q) = cos(90°−lat₁)cos(90°−lat₂) + sin(90°−lat₁)sin(90°−lat₂)cos(Δlon)

As above, simplifies to:

cos(q) = sin(lat₁)sin(lat₂) + cos(lat₁)cos(lat₂)cos(Δlon)

Therefore, the angular distance q is:

q = arccos[ sin(lat₁)sin(lat₂) + cos(lat₁)cos(lat₂)cos(Δlon) ]

Then physical distance = R × q (with q in radians)

But again — without numbers, we can’t compute a numeric answer.

Wait — perhaps the problem expects us to recognize that the green arc labeled “Great-circle distance q” is computed via that formula?

Maybe the “task” is simply to identify how to calculate q from the given quantities?

Given that, and since no numbers are provided, I believe the correct response is to present the formula derived from the diagram.

But let’s double-check if there's any hidden info.

The diagram also mentions “Greenwich meridian” and “Equator”, but those are just reference lines.

Also note: latitudes go from -90° to 90°, longitudes from -180° to 180°.

Important: When computing, convert degrees to radians first!

Formula summary:

Step 1: Convert lat₁, lat₂, Δlon from degrees to radians.

Step 2: Compute:

cos_q = sin(lat₁_rad) * sin(lat₂_rad) + cos(lat₁_rad) * cos(lat₂_rad) * cos(Δlon_rad)

Step 3: q_rad = arccos(cos_q)

Step 4: Distance = R * q_rad

Since R isn’t specified, leave it as R, or assume Earth’s radius if context implies.

But again — no numbers.

Perhaps the original problem had specific coordinates? For example, common textbook problems use:

P₁: (40°N, 75°W)
P₂: (35°N, 120°E) etc.

But since none are given, I think the best I can do is provide the general solution method.

Wait — maybe the question is implied to be: “What is the formula for great-circle distance based on this diagram?”

In that case, the answer is:

q = arccos[ sin(lat₁) sin(lat₂) + cos(lat₁) cos(lat₂) cos(lon₂ - lon₁) ]

With angles in radians.

To make sure, let’s verify with a known case.

Example: Same point — lat₁=lat₂, lon₁=lon₂ → Δlon=0

Then cos(q) = sin²(lat) + cos²(lat)*1 = sin²(lat) + cos²(lat) = 1 → q=0 ✓

Another: Equator, same longitude — lat₁=lat₂=0, Δlon=θ

cos(q) = 0*0 + 1*1*cos(θ) = cos(θ) → q=θ ✓

Good.

So the formula is correct.

Since no numerical values are provided in the query, and the image description only contains symbols, I conclude that the expected answer is the formula itself.

Final Answer:
The great-circle distance \( q \) (in radians) between two points \( P_1(\text{lat}_1, \text{lon}_1) \) and \( P_2(\text{lat}_2, \text{lon}_2) \) on a sphere is given by:

\[
q = \arccos\left( \sin(\text{lat}_1) \sin(\text{lat}_2) + \cos(\text{lat}_1) \cos(\text{lat}_2) \cos(\Delta\text{lon}) \right)
\]

where \( \Delta\text{lon} = |\text{lon}_1 - \text{lon}_2| \), and all angles must be in radians. The actual distance is then \( d = R \cdot q \), where \( R \) is the sphere’s radius.