# 9.5 Solving trigonometric equations

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In this section, you will:
• Solve linear trigonometric equations in sine and cosine.
• Solve equations involving a single trigonometric function.
• Solve trigonometric equations using a calculator.
• Solve trigonometric equations that are quadratic in form.
• Solve trigonometric equations using fundamental identities.
• Solve trigonometric equations with multiple angles.
• Solve right triangle problems.

Thales of Miletus (circa 625–547 BC) is known as the founder of geometry. 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. Based on proportions, this theory has applications in a number of areas, including fractal geometry, engineering, and architecture. Often, the angle of elevation and the angle of depression are found using similar triangles.

In earlier sections of this chapter, we looked at trigonometric identities. Identities are true for all values in the domain of the variable. In this section, we begin our study of trigonometric equations to study real-world scenarios such as the finding the dimensions of the pyramids.

## Solving linear trigonometric equations in sine and cosine

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 $\text{\hspace{0.17em}}2\pi .$ In other words, every $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ units, the y- values repeat. If we need to find all possible solutions, then we must add $\text{\hspace{0.17em}}2\pi k,$ where $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}2\pi \text{:}$

$\mathrm{sin}\text{\hspace{0.17em}}\theta =\mathrm{sin}\left(\theta ±2k\pi \right)$

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.

## Solving a linear trigonometric equation involving the cosine function

Find all possible exact solutions for the equation $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =\frac{1}{2}.$

From the unit circle    , we know that

$\begin{array}{ccc}\hfill \mathrm{cos}\text{\hspace{0.17em}}\theta & =& \frac{1}{2}\hfill \\ \hfill \theta & =& \frac{\pi }{3},\frac{5\pi }{3}\hfill \end{array}$

These are the solutions in the interval $\text{\hspace{0.17em}}\left[0,2\pi \right].\text{\hspace{0.17em}}$ All possible solutions are given by

where $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is an integer.

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