# Trigonometric function introduction

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## Introduction

Trigonometry is a very important tool for engineers. This reading mentions just a few of the many applications that involve trigonometry to solve engineering problems.

Trigonometry means the study of the triangle. Most often, it refers to finding angles of a triangle when the lengths of the sides are known, or finding the lengths of two sides when the angles and one of the side lengths are known.

As you complete this reading, be sure to pay special attention to the variety of areas in which engineers utilize trigonometry to develop the solutions to problems. Rest assured that there are an untold number of applications of trigonometry in engineering. This reading only introduces you to a few. You will learn many more as you progress through your engineering studies.

Virtually all engineers use trigonometry in their work on a regular basis. Things like the generation of electrical current or a computer use angles in ways that are difficult to see directly, but that rely on the fundamental rules of trigonometry to work properly. Any time angles appear in a problem, the use of trigonometry usually will not be far behind.

An excellent working knowledge of trigonometry is essential for practicing engineers. Many problems can be easily solved by applying the fundamental definitions of trigonometric functions. Figure 1 depicts a right triangle for which we will express the various trigonometric functions and relationships that are important to engineers.

Some of the most widely used trigonometric functions follow

$\text{sin}\left(\theta \right)=\frac{\text{opposite}\text{side}}{\text{hypotenuse}}=\frac{B}{C}$
$\text{cos}\left(\theta \right)=\frac{\text{adjacent}\text{side}}{\text{hypotenuse}}=\frac{A}{C}$
$\text{tan}\left(\theta \right)=\frac{\text{opposite}\text{side}}{\text{adjacent}\text{side}}=\frac{B}{A}$

The Pythagorean Theorem often plays a key role in applications involving trigonometry in engineering. It states that the square of the hypotenuse is equal to the sum of the squares of the adjacent side and of the opposite side. This theorem can be stated mathematically by means of the equation that follows. Here we make use of the symbols (A, B, and C) that designate associated lengths.

${A}^{2}+{B}^{2}={C}^{2}$

In the remainder of this module, we will make use of these formulas in addressing various applications.

## Flight path of an aircraft

The following is representative of the type of problem one is likely to encounter in the introductory Physics course as well as in courses in Mechanical Engineering. This represents an example of how trigonometry can be applied to determine the motion of an aircraft.

Assume that an airplane climbs at a constant angle of 3 0 from its departure point situated at sea level. It continues to climb at this angle until it reaches its cruise altitude. Suppose that its cruise altitude is 31,680 ft above sea level.

With the information stated above, we may sketch an illustration of the situation

Question 1: What is the distance traveled by the plane from its departure point to its cruise altitude?

The distance traveled by the plane is equal to the length of the hypotenuse of the right triangle depicted in Figure 2. Let us denote the distance measured in feet that the plane travels by the symbol C . We may apply the definition of the sine function to enable us to solve for the symbol C as follows.

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