Dallas is 190 miles due south of Oklahoma City, and Fort...

Dallas is 190 miles due south of Oklahoma City, and Fort Smith is 140 miles due east of Oklahoma City. An airplane flies on a direct trip from Dallas to Fort Smith. a. What is the tangent of the angle that the flight path makes with Interstate 35 , which runs due north from Dallas to Oklahoma City? b. What is the angle that the flight path makes with Interstate 35 ? c. How far does the airplane fly?

Dallas is 190 miles due south of Oklahoma City, and Fort Smith is 140 miles due east of Oklahoma City. An airplane flies on a direct trip from Dallas to Fort Smith.
a. What is the tangent of the angle that the flight path makes with Interstate 35 , which runs due north from Dallas to Oklahoma City?
b. What is the angle that the flight path makes with Interstate 35 ?
c. How far does the airplane fly?

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Key Concepts

Right Triangle Trigonometry

This concept involves analyzing right triangles formed by the given distances and angles. It allows the use of side length ratios to determine unknown angles or sides, making it essential in solving navigation, surveying, and other problems that involve spatial relations.

Tangent Function

The tangent function in trigonometry is the ratio of the length of the side opposite an angle to the length of the adjacent side in a right triangle. This function is used to determine the steepness of an angle and is central to calculating angular measurements in right triangle problems.

Inverse Tangent

The inverse tangent function, often written as arctan, is used to determine the measure of an angle when the tangent value is known. It effectively reverses the tangent operation, making it essential for finding angle measures from side ratios in right triangle scenarios.

Pythagorean Theorem

The Pythagorean Theorem establishes the relationship between the sides of a right triangle, stating that the sum of the squares of the legs is equal to the square of the hypotenuse. This theorem is critical for calculating distances, such as the direct flight path between two points, when the horizontal and vertical displacements are known.

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