To solve the equation [tex]\( 5 \sin(2 \theta) - 6 \sin(\theta) = 0 \)[/tex], we need to proceed through a series of steps to find the values of [tex]\(\theta\)[/tex]. Here's a step-by-step guide:
1. Express [tex]\(\sin(2 \theta)\)[/tex] in terms of [tex]\(\sin(\theta)\)[/tex] and [tex]\(\cos(\theta)\)[/tex]:
Recall that [tex]\(\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)\)[/tex]. Substitute this into the equation:
[tex]\[
5 \cdot 2 \sin(\theta) \cos(\theta) - 6 \sin(\theta) = 0
\][/tex]
Simplify it to:
[tex]\[
10 \sin(\theta) \cos(\theta) - 6 \sin(\theta) = 0
\][/tex]
2. Factor out the common term:
Notice that [tex]\(\sin(\theta)\)[/tex] is common in both terms, so factor it out:
[tex]\[
\sin(\theta) \left( 10 \cos(\theta) - 6 \right) = 0
\][/tex]
3. Set each factor to zero and solve:
For [tex]\(\sin(\theta) = 0\)[/tex]:
[tex]\[
\theta = 0 + k \pi \quad \text{(where } k \text{ is any integer)}
\][/tex]
Typically, we select the fundamental solutions which are [tex]\( \theta = 0 \)[/tex] and [tex]\( \theta = \pi \)[/tex]. These are straightforward solutions in the real number domain.
4. Solve for the second factor [tex]\(10 \cos(\theta) - 6 = 0\)[/tex]:
[tex]\[
10 \cos(\theta) = 6
\][/tex]
[tex]\[
\cos(\theta) = \frac{6}{10} = \frac{3}{5}
\][/tex]
To find [tex]\(\theta\)[/tex] where [tex]\(\cos(\theta) = \frac{3}{5}\)[/tex]:
For real solutions of [tex]\(\theta\)[/tex], there are no principal angles that yield [tex]\(\cos(\theta) = \frac{3}{5}\)[/tex] directly since we are considering real angles. However, solving this equation in the complex domain may introduce complex solutions.
5. Consider the complex solutions:
Solve:
[tex]\(\cos(\theta) = \frac{3}{5}\)[/tex]
Remember the general form [tex]\(\theta = \pm \cos^{-1}\left(\frac{3}{5}\right) + 2k\pi\)[/tex], which gives complex solutions if not restricted to real [tex]\(k\)[/tex].
6. Compile all solutions:
The real solutions are:
[tex]\[
\theta = 0, \pi
\][/tex]
In addition, the complex solutions can be expressed, which after solving under the complex domain, yield:
[tex]\[
\theta = i \left( \ln(5) - \ln(3 - 4i) \right) \quad \text{and} \quad \theta = i \left( \ln(5) - \ln(3 + 4i) \right)
\][/tex]
Thus, the complete set of solutions to the equation [tex]\(5 \sin(2\theta) - 6 \sin(\theta) = 0\)[/tex] are:
[tex]\[
\theta = 0, \pi, i \left( \ln(5) - \ln(3 - 4i) \right), i \left( \ln(5) - \ln(3 + 4i) \right)
\][/tex]
