Problems involving the reciprocals of the primary trigonometric functions need to be viewed from an algebraic perspective. We need to make several considerations when the equation involves trigonometric functions other than sine and cosine. When we are given equations that involve only one of the six trigonometric functions, their solutions involve using algebraic techniques and the unit circle (see Figure 2). Solving Equations Involving a Single Trigonometric Function Solve exactly the following linear equation on the interval [ 0, 2 π ) : 2 sin x + 1 = 0. Recall the rule that gives the format for stating all possible solutions for a function where the period is 2 π : 2 π : If we need to find all possible solutions, then we must add 2 π k, 2 π k, where k k is an integer, to the initial solution. In other words, every 2 π 2 π units, the y-values repeat. The period of both the sine function and the cosine function is 2 π. Additionally, like rational equations, the domain of the function must be considered before we assume that any solution is valid. In other words, trigonometric equations may have an infinite number of solutions. However, just as often, we will be asked to find all possible solutions, and as trigonometric functions are periodic, solutions are repeated within each period. Often we will solve a trigonometric equation over a specified interval.

Similar in many ways to solving polynomial equations or rational equations, only specific values of the variable will be solutions, if there are solutions at all. Trigonometric equations are, as the name implies, equations that involve trigonometric functions. Solving Linear Trigonometric Equations in Sine and Cosine In this section, we begin our study of trigonometric equations to study real-world scenarios such as the finding the dimensions of the pyramids. Identities are true for all values in the domain of the variable. In earlier sections of this chapter, we looked at trigonometric identities. Often, the angle of elevation and the angle of depression are found using similar triangles. Based on proportions, this theory has applications in a number of areas, including fractal geometry, engineering, and architecture. The legend is that he calculated the height of the Great Pyramid of Giza in Egypt using the theory of similar triangles, which he developed by measuring the shadow of his staff.

Thales of Miletus (circa 625–547 BC) is known as the founder of geometry. Figure 1 Egyptian pyramids standing near a modern city.

Sine function equation maker