SOLVING TRIGONOMETRIC EQUATIONS

Note:

If you would like an review of trigonometry, click on trigonometry.

Solve for x in the following equation.

Example 2:        

$2sin\left( x\right) -\displaystyle \frac{3}{5}=0$

There are an infinite number of solutions to this problem. To solve for x, isolate the sine term.

$$\begin{eqnarray*}&& \\ 2sin\left( x\right) -\displaystyle \frac{3}{5}=0 && \\ ... ... && \\ && \\ sin\left( x\right) =\displaystyle \frac{3}{10} && \end{eqnarray*}$$

$$\begin{eqnarray*}&& \\ If\ we\ restrict\ the\ domain\ of\ the\ function\ to\ \l... ...\ && \\ && \\ x\approx 0.304692654015 && \\ && \\ && \\ && \end{eqnarray*}$$

The sine function is positive in the first and second quadrant. If the reference angle is $x\approx 0.304692654015$, the angle that terminates in the second quadrant is $\pi -0.304692654015=2.83689999.$

The period of sin $\left( x\right)$ function is $2\pi .$ This means that the values will repeat every $2\pi$ radians. Therefore, the solutions are $x=0.304692654015\pm n\left( 2\pi \right)$ and $x=2.836899999\pm n\left( 2\pi \right)$ where n is an integer.

These solutions may or may not be the answers to the original problem. You much check them, either numerically or graphically, with the original equation.

Numerical Check:

Check answer . x=0.304692654015

• Left Side: $\qquad 2sin\left( 0.304692654015\right) -\displaystyle \frac{3}{5}\approx 0$

• Right Side:        0

Since the left side equals the right side when you substitute 0.304692654015 for x, then 0.304692654015 is a solution.

Check answer . x=2.83689999957

• Left Side: $\qquad 2sin\left( 2.83689999957\right) -\displaystyle \frac{3}{5}\approx 0$

• Right Side:        0

Since the left side equals the right side when you substitute 2.83689999957for x, then 2.83689999957 is a solution.

Graphical Check:

Graph the equation

$f(x)=2\sin (x)-\displaystyle \frac{3}{5}.$

Note that the graph crosses the x-axis many times indicating many solutions.

Note that it crosses at (one of the solutions). Since the period of the function is $2\pi \approx 6.28318530718$, the graph crosses again at 2.83689999957+6.28318530718=9.12 and again at $2.83689999957+2\left( 6.28318530718\right) \approx 15.4$, etc. The graph also crosses at 0.304692654015 (another solution we found). Since the period is $2\pi \approx 6.28318530718$, it will cross again at $0.304692654015+2\pi =6.58788$ and at $0.304692654015+2\left( 2\pi \right) \approx 12.871$, etc.

If you would like to work another example, click on Example.

If you would like to test yourself by working some problems similar to this example, click on Problem.

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