Alright, let’s solve the trigonometric equation [tex]\(\sin 3\theta = \cos 6\theta\)[/tex].
First, recall a useful trigonometric identity that can help to simplify our equation:
[tex]\[ \cos 6\theta = \sin(90^\circ - 6\theta) \][/tex]
Therefore, we can rewrite our original equation as:
[tex]\[ \sin 3\theta = \sin(90^\circ - 6\theta) \][/tex]
For two sine functions to be equal, their arguments must differ by an integer multiple of [tex]\(360^\circ\)[/tex] (considering the periodicity of the sine function) or be supplementary angles. This means we have two potential sets of solutions:
1. The arguments are directly equal plus a multiple of [tex]\(360^\circ\)[/tex]:
[tex]\[ 3\theta = 90^\circ - 6\theta + 360^\circ k \][/tex]
where [tex]\(k\)[/tex] is any integer.
2. The angles are supplementary (since [tex]\(\sin x = \sin(180^\circ - x)\)[/tex]) plus a multiple of [tex]\(360^\circ\)[/tex]:
[tex]\[ 3\theta = 180^\circ - (90^\circ - 6\theta) + 360^\circ k \][/tex]
where [tex]\(k\)[/tex] is any integer.
### Solving the First Equation:
Starting with:
[tex]\[ 3\theta = 90^\circ - 6\theta + 360^\circ k \][/tex]
Combine like terms:
[tex]\[ 3\theta + 6\theta = 90^\circ + 360^\circ k \][/tex]
[tex]\[ 9\theta = 90^\circ + 360^\circ k \][/tex]
Divide by 9:
[tex]\[ \theta = \frac{90^\circ + 360^\circ k}{9} \][/tex]
[tex]\[ \theta = 10^\circ + 40^\circ k \][/tex]
### Solving the Second Equation:
Starting with:
[tex]\[ 3\theta = 180^\circ - (90^\circ - 6\theta) + 360^\circ k \][/tex]
[tex]\[ 3\theta = 180^\circ - 90^\circ + 6\theta + 360^\circ k \][/tex]
[tex]\[ 3\theta = 90^\circ + 6\theta + 360^\circ k \][/tex]
Combine like terms:
[tex]\[ 3\theta - 6\theta = 90^\circ + 360^\circ k \][/tex]
[tex]\[ -3\theta = 90^\circ + 360^\circ k \][/tex]
Divide by -3:
[tex]\[ \theta = -30^\circ - 120^\circ k \][/tex]
### General Solutions:
Thus, the solutions to the equation [tex]\(\sin 3\theta = \cos 6\theta\)[/tex] are given by:
[tex]\[ \theta = 10^\circ + 40^\circ k \][/tex]
and
[tex]\[ \theta = -30^\circ - 120^\circ k \][/tex]
For a comprehensive set of solutions, we might list them considering a specific range for [tex]\(k\)[/tex]. For instance, if we consider [tex]\(k\)[/tex] in the range from [tex]\(-10\)[/tex] to [tex]\(10\)[/tex], we get the respective solutions:
First set of solutions:
[tex]\[
[-390.0^\circ, -350.0^\circ, -310.0^\circ, -270.0^\circ, -230.0^\circ, -190.0^\circ, -150.0^\circ, -110.0^\circ, -70.0^\circ, -30.0^\circ, 10.0^\circ, 50.0^\circ, 90.0^\circ, 130.0^\circ, 170.0^\circ, 210.0^\circ, 250.0^\circ, 290.0^\circ, 330.0^\circ, 370.0^\circ, 410.0^\circ]
\][/tex]
Second set of solutions:
[tex]\[
[-390.0^\circ, -350.0^\circ, -310.0^\circ, -270.0^\circ, -230.0^\circ, -190.0^\circ, -150.0^\circ, -110.0^\circ, -70.0^\circ, -30.0^\circ, 10.0^\circ, 50.0^\circ, 90.0^\circ, 130.0^\circ, 170.0^\circ, 210.0^\circ, 250.0^\circ, 290.0^\circ, 330.0^\circ, 370.0^\circ, 410.0^\circ]
\][/tex]
Note that for this particular example, both sets of solutions overlap. These values represent solutions where the given equation [tex]\(\sin 3\theta = \cos 6\theta\)[/tex] holds true.
