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Solving Trigonometric Equations

A trigonometric equation is an equation that contains trigonometric functions such as sine and cosine. There are usually two ways to solve a trigonometric equation, an algebraic method and a graphical method. When solving algebraically, we first isolate the trigonometric function and then use the inverse of that function to solve for the variable. Since the trigonometric functions are periodic, a trigonometric equation can have infinitely many solutions. Depending on the situation, we may be asked to solve the equation over a specified interval or we may be asked to find all solutions of the equation. When we are asked to find all solutions of a trigonometric equation, we first find the solutions over the interval representing one complete cycle of the graph, and then use the fact that the trigonometric functions are periodic to find the remaining solutions.

Example 1

Find all solutions of the equation .


The first step is to solve for .


One solution to this equation is . While the inverse sine gives only one solution, there are actually an infinite number of solutions. Since the sine function has period 2p, we should first find all solutions in the interval . Then we can add any integer multiple of 2p to these solutions to get another solution. To find all solutions in the interval , we can think of t as the radian measure of an angle in standard position whose terminal side intersects the unit circle at a point with y-coordinate . There are exactly two such angles, one in Quadrant I, and the other in Quadrant II. See Figure 5.

     Figure 5

We already know that the angle in quadrant I has radian measure  Using the symmetry of the circle, we can easily determine that the angle in Quadrant II is  Finally, we see that all solutions are of the form,


We can also write the solutions as decimals,


Example 2

Find all solutions to the equation  over the interval .


We first solve for cos t,

 The inverse cosine function gives  Since the cosine function is also negative in Quadrant III, there is another solution in that quadrant. It is (rounded to four decimal places). See Figure 6.

      Figure 6