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Ch. 3 - Radian Measure and The Unit Circle
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 3 - Radian Measure and The Unit CircleProblem 26
Chapter 4, Problem 26

Distance between Cities Find the distance in kilometers between each pair of cities, assuming they lie on the same north-south line. Assume the radius of Earth is 6400 km. See Example 2.
Halifax, Nova Scotia , 45° N, and Buenos Aires, Argentina, 34° S

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1
Identify the latitudes of the two cities: Halifax is at 45° N and Buenos Aires is at 34° S. Since they are on opposite sides of the equator, the total angular distance between them is the sum of their absolute latitudes.
Calculate the total central angle \( \theta \) between the two cities on the Earth's surface by adding the absolute values of their latitudes: \( \theta = 45^\circ + 34^\circ \).
Convert the central angle \( \theta \) from degrees to radians because the arc length formula requires the angle in radians. Use the conversion formula: \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \).
Use the arc length formula to find the distance \( d \) between the two cities along the Earth's surface: \[ d = R \times \theta_{radians} \] where \( R = 6400 \) km is the radius of the Earth.
Substitute the values of \( R \) and \( \theta_{radians} \) into the formula to express the distance \( d \) in kilometers. This will give the distance between Halifax and Buenos Aires along the north-south line.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Great Circle Distance on a Sphere

The shortest distance between two points on the surface of a sphere lies along the great circle connecting them. When cities lie on the same meridian (north-south line), the distance can be found by calculating the arc length between their latitudes on the Earth's surface.
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Latitude and Angular Distance

Latitude measures the angular distance north or south of the equator. To find the distance between two points on the same meridian, subtract their latitudes (considering north and south as positive and negative) to get the central angle in degrees, which is then converted to radians for calculations.
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Example 1

Arc Length Formula on a Sphere

The arc length between two points on a sphere is given by s = rθ, where r is the radius of the sphere and θ is the central angle in radians. This formula allows conversion of angular separation into a linear distance along the Earth's surface.
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Related Practice
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Textbook Question

Use the formula v = r ω to find the value of the missing variable.


v = 12 m per sec, ω = 3π/2 radians per sec

Textbook Question

The formula ω = θ/t can be rewritten as θ = ωt. Substituting ωt for θ converts s = rθ to s = rωt. Use the formula s = rωt to find the value of the missing variable.


r = 6 cm, ω = π/3 radians per sec, t = 9 sec