To find the value of [tex]\(\tan \theta\)[/tex] given the equation [tex]\(8 \sin \theta = 4 + \cos \theta\)[/tex], we will follow these steps:
1. Express [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] in a manageable form:
Let's denote:
[tex]\[
s = \sin \theta \quad \text{and} \quad c = \cos \theta
\][/tex]
This transforms our given equation into:
[tex]\[
8s = 4 + c
\][/tex]
2. Utilize the Pythagorean identity:
We know the fundamental trigonometric identity:
[tex]\[
s^2 + c^2 = 1
\][/tex]
3. Substitute and solve the system of equations:
From the first equation [tex]\(8s = 4 + c\)[/tex], we can solve for [tex]\(c\)[/tex]:
[tex]\[
c = 8s - 4
\][/tex]
Substitute this expression for [tex]\(c\)[/tex] into the Pythagorean identity:
[tex]\[
s^2 + (8s - 4)^2 = 1
\][/tex]
4. Simplify and solve the quadratic equation:
Expanding and simplifying the equation:
[tex]\[
s^2 + (64s^2 - 64s + 16) = 1
\][/tex]
[tex]\[
s^2 + 64s^2 - 64s + 16 = 1
\][/tex]
[tex]\[
65s^2 - 64s + 15 = 0
\][/tex]
To solve this quadratic equation, we use the quadratic formula [tex]\(s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 65\)[/tex], [tex]\(b = -64\)[/tex], and [tex]\(c = 15\)[/tex].
Solving this, we find two potential values for [tex]\(s\)[/tex]:
[tex]\[
s_1 = 0.384615384615385 \quad \text{and} \quad s_2 = \text{another potential solution}
\][/tex]
5. Find the corresponding values of [tex]\(\cos \theta\)[/tex] for valid [tex]\(\sin \theta\)[/tex]:
Using the first value [tex]\(s_1 = 0.384615384615385\)[/tex]:
[tex]\[
c_1 = 8s_1 - 4 \approx -0.923076923076923
\][/tex]
6. Calculate [tex]\(\tan \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
Substitute the values:
[tex]\[
\tan \theta = \frac{0.384615384615385}{-0.923076923076923} \approx -0.416666666666667
\][/tex]
Hence, the value of [tex]\(\tan \theta\)[/tex] is approximately [tex]\(-0.416666666666667\)[/tex].
