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Sagot :
Trigonometric Equations
Solve:
[tex]cos(\pi\theta)=-0.6[/tex]Using the inverse cosine function in the calculator, we get:
[tex]\begin{gathered} \pi\theta=\arccos(-0.6) \\ \\ \pi\theta=2.2143\text{ rad} \end{gathered}[/tex]There is another solution in quadrant III
[tex]\begin{gathered} \\ \pi\theta=(2\pi-2.2143)\text{ rad} \\ \\ \pi\theta=4.069\text{ rad} \end{gathered}[/tex]Dividing by π, we get the first two solutions:
[tex]\begin{gathered} \\ \theta=\frac{2.2143}{\pi}\text{ rad} \\ \\ \theta=0.7048\text{ rad} \end{gathered}[/tex]The second solution is:
[tex]\begin{gathered} \theta=\frac{4.069}{\pi}\text{ rad} \\ \\ \theta=1.2952\text{ rad} \end{gathered}[/tex]We can find more solutions by adding 2π to the inverse cosine angle.
[tex]\begin{gathered} \pi\theta=(2.2143+2\pi)\text{ rad} \\ \\ \pi\theta=8.4975\text{ rad} \end{gathered}[/tex]Dividing by π:
[tex]\begin{gathered} \theta=\frac{8.4975}{\pi}\text{ rad} \\ \\ \theta=2.7078\text{ rad} \end{gathered}[/tex]A fourth solution is found as follows:
[tex]\begin{gathered} \pi\theta=(2\pi+4.069)\text{ rad} \\ \\ \theta=\frac{10.3521}{\pi}\text{ rad} \\ \\ \theta=3.2952\text{ rad} \end{gathered}[/tex]We must keep adding 2π until the solution goes outside of the interval (0, 2π).
[tex]\begin{gathered} \pi\theta=(8.4975+2\pi)\text{ rad} \\ \\ \pi\theta=14.7807\text{ rad} \\ \\ \theta=\frac{14.7807}{\pi}\text{rad} \\ \\ \theta=4.7048\text{ rad} \end{gathered}[/tex]The final solution is found as follows:
[tex]\begin{gathered} \pi\theta=(2\pi+10.3521)\text{ rad} \\ \\ \theta=\frac{16.6353}{\pi}\text{ rad} \\ \\ \theta=5.2952\text{ rad} \end{gathered}[/tex]The complete set of solutions is given below:
θ = 0.70 rad
θ = 1.30 rad
θ = 2.70 rad
θ = 3.30 rad
θ = 4.70 rad
θ = 5.30 rad
The next solution would be 4.70 rad but it's greater than 2π, so we stop here
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