how can you tell how many solutions you'll get for a trigonometric equations? does it depend on the value of theta or the range?

Trigonometric equations are, as the name implies, equations that involve trigonometric functions. 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. Often we will solve a trigonometric equation over a specified interval. 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. In other words, trigonometric equations may have an infinite number of solutions. Additionally, like rational equations, the domain of the function must be considered before we assume that any solution is valid. The period of both the sine function and the cosine function is \displaystyle 2\pi2π. In other words, every \displaystyle 2\pi2π units, the y-values repeat. If we need to find all possible solutions, then we must add \displaystyle 2\pi k2πk, where \displaystyle kk is an integer, to the initial solution. Recall the rule that gives the format for stating all possible solutions for a function where the period is \displaystyle 2\pi :2π:

\displaystyle \sin \theta =\sin \left(\theta \pm 2k\pi \right)sinθ=sin(θ±2kπ)

There are similar rules for indicating all possible solutions for the other trigonometric functions. Solving trigonometric equations requires the same techniques as solving algebraic equations. We read the equation from left to right, horizontally, like a sentence. We look for known patterns, factor, find common denominators, and substitute certain expressions with a variable to make solving a more straightforward process. However, with trigonometric equations, we also have the advantage of using the identities we developed in the previous sections.

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