Ch. 3 - Radian Measure and The Unit Circle

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 3 - Radian Measure and The Unit Circle

Problema 23

Capitolo 4, Problema 23

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. Panama City, Panama, 9° N, and Pittsburgh, Pennsylvania, 40° N

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Identify the latitudes of the two cities: Panama City at 9° N and Pittsburgh at 40° N. Since they lie on the same north-south line, the difference in latitude will determine the distance between them.

Calculate the difference in latitude between the two cities: \(\Delta \theta = 40^\circ - 9^\circ\).

Convert the difference in latitude from degrees to radians because the arc length formula requires the angle in radians. Use the conversion formula: \(\text{radians} = \Delta \theta \times \frac{\pi}{180}\).

Use the arc length formula to find the distance along the Earth's surface: \(\text{distance} = R \times \text{radians}\), where \(R = 6400\) km is the radius of the Earth.

Substitute the values into the formula and simplify to express the distance between Panama City and Pittsburgh in kilometers.

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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. For points on the same meridian (north-south line), this distance corresponds to the arc length between their latitudes on the Earth's surface.

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Percorso guidato

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Introduction to the Unit Circle

Arc Length Calculation Using Central Angle

The distance between two points on a circle can be found by multiplying the radius by the central angle (in radians) between them. Here, the central angle is the difference in latitude degrees converted to radians, and the radius is the Earth's radius.

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How to Use a Calculator for Trig Functions

Conversion Between Degrees and Radians

Angles measured in degrees must be converted to radians for use in arc length formulas. The conversion is done by multiplying degrees by π/180, since 180 degrees equals π radians.

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Converting between Degrees & Radians

Pratica correlata

Domanda del libro di testo

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.

Farmersville, California, 36° N, and Penticton, British Columbia, 49° N

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Domanda del libro di testo

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.

New York City, New York, 41° N, and Lima, Peru, 12° S

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Domanda del libro di testo

Convert each degree measure to radians. Leave answers as multiples of π. See Examples 1(a) and 1(b). ―900°

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Domanda del libro di testo

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

v = 9 m per sec , r = 5 m

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