A car is driven 13 mi east and then a certain distance due...

A car is driven 13 mi east and then a certain distance due north, ending up at a position $25^{\circ}$ north of east of its initial position. (a) The distance traveled by the car due north is (1) less than, (2) equal to, (3) greater than 13 mi. Why? (b) What distance due north does the car travel?

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Key Concepts

Trigonometric Ratios (Tangent Function)

The tangent function, defined as the ratio of the opposite side to the adjacent side in a right triangle, provides a direct means of comparing the two perpendicular components of a displacement vector. When the angle the resultant vector makes with one axis is known, the tangent ratio can be used to find the unknown component relative to the known one, and also to compare the relative sizes of these components.

Vector Decomposition

This concept involves breaking down a movement or displacement into its component parts along mutually perpendicular directions, typically north–south and east–west. It is used to analyze overall movement by separately considering the contributions of each directional component, which can then be added together to obtain the resultant displacement.

Right Triangle Trigonometry

In right triangle trigonometry, the relationships among the angles and sides of a right triangle are used to solve for unknown lengths or angles. This is essential when the overall displacement and one component are known, allowing one to determine the other component using trigonometric functions like sine, cosine, and tangent.

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