SOLVING TRIGONOMETRIC EQUATIONS

Note:

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

Solve for x in the following equation.

Example 1:         

$2sin\left( x\right) -1=0$

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

$$\begin{eqnarray*}&& \\ 2\sin \left( x\right) -1 &=&0 \\ && \\ \sin \left( x\right) &=&\displaystyle \frac{1}{2} \\ && \\ && \end{eqnarray*}$$

We know that the $\sin \left( \displaystyle \frac{\pi }{6}\right) =\displaystyle \frac{1}{2},$therefore $x=\displaystyle \frac{\pi }{6}.$ The sine function is positive in quadrants I and II. The $\sin \left( \pi -\displaystyle \frac{\pi }{6}\right) =\sin \displaystyle \frac{5\pi }{6}$is also equal to $$\frac{1}{2}.$$ Therefore, two of the solutions to the problem are $x=\displaystyle \frac{\pi }{6}$ and $x=\displaystyle \frac{5\pi }{6}.$

The period of the sin $\left( x\right)$ function is $2\pi .$ This means that the values will repeat every $2\pi$ radians in both directions. Therefore, the exact solutions are $x=\displaystyle \frac{\pi }{6}\pm n\left( 2\pi \right)$ and $x= \displaystyle \frac{5\pi }{6}\pm n\left( 2\pi \right)$ where n is an integer. The approximate solutions are $x=0.523598775598\pm n\left( 2\pi \right)$ and $x=2.61799387799\pm n\left( 2\pi \right)$ 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 . $x=\displaystyle \frac{\pi }{6}$

• Left Side: $\qquad 2\sin \left( x\right) -1=2\sin \left( \displaystyle \frac{\pi }{6} \right) -1=2\left( \displaystyle \frac{1}{2}\right) -1=0$

• Right Side:        0

Since the left side equals the right side when you substitute $$\frac{\pi }{6 }$$ for x, then $$\frac{\pi }{6 }$$ is a solution.

Check answer . $x=\displaystyle \frac{5\pi }{6}$

• Left Side: $\qquad 2\sin \left( x\right) -1=2\sin \left( \displaystyle \frac{5\pi }{6} \right) -1=2\left( \displaystyle \frac{1}{2}\right) -1=0$

• Right Side:        0

Since the left side equals the right side when you substitute $$\frac{5\pi }{ 6}$$ for x, then $$\frac{5\pi }{ 6}$$ is a solution.

Graphical Check:

Graph the equation

$f(x)=2\sin x(x)-1.$

Note that the graph crosses the x-axis many times indicating many solutions. Note that it crosses at $$\frac{ \pi }{6}\approx 0.5236$$. Since the period is $2\pi \approx 6.2831$, it crosses again at 0.5236+6.283=6.81 and at 0.5236+2(6.283)=13.09, etc. The graph crosses at $$\frac{5\pi }{6}\approx 2.618$$. Since the period is $2\pi \approx 6.283$, it will crosses again at 2.618+6.283=8.9011 and at 2.618+2(6.283)=15.18, 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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