To solve the equation [tex]\(\sec(4\theta) = \operatorname{cosec}(\theta - 300^\circ)\)[/tex] given that [tex]\(4\theta\)[/tex] is an acute angle, follow these steps:
1. Convert the trigonometric identities:
We know that [tex]\(\sec(x) = \frac{1}{\cos(x)}\)[/tex] and [tex]\(\operatorname{cosec}(x) = \frac{1}{\sin(x)}\)[/tex].
2. Rewrite the equation using these identities:
[tex]\[
\frac{1}{\cos(4\theta)} = \frac{1}{\sin(\theta - 300^\circ)}
\][/tex]
3. Equate the denominators, since the numerators (1) are equal:
[tex]\[
\cos(4\theta) = \sin(\theta - 300^\circ)
\][/tex]
4. Use the complementary angle identity of trigonometric functions:
We know that [tex]\(\sin(x) = \cos(90^\circ - x)\)[/tex].
Thus, [tex]\(\sin(\theta - 300^\circ) = \cos\left(90^\circ - (\theta - 300^\circ)\right)\)[/tex].
Simplify inside the parenthesis:
[tex]\[
\cos\left(90^\circ - (\theta - 300^\circ)\right) = \cos\left(90^\circ - \theta + 300^\circ\right) = \cos(390^\circ - \theta)
\][/tex]
5. Substitute this back into the equation:
[tex]\[
\cos(4\theta) = \cos(390^\circ - \theta)
\][/tex]
6. Since [tex]\(\cos(x) = \cos(y)\)[/tex] implies [tex]\(x = y + 360^\circ k\)[/tex] or [tex]\(x = -y + 360^\circ k\)[/tex] (for any integer [tex]\(k\)[/tex]), we solve for [tex]\(\theta\)[/tex]:
[tex]\(4\theta = 390^\circ - \theta + 360^\circ k\)[/tex] or [tex]\(4\theta = - 390^\circ + \theta + 360^\circ k\)[/tex].
Let’s focus on the positive cycle only, i.e., [tex]\(k = 0\)[/tex], to find valid angles within the given range.
For the first equation:
[tex]\[
4\theta = 390^\circ - \theta
\][/tex]
Add [tex]\(\theta\)[/tex] to both sides:
[tex]\[
5\theta = 390^\circ
\][/tex]
Solve for [tex]\(\theta\)[/tex]:
[tex]\[
\theta = \frac{390^\circ}{5} = 78^\circ
\][/tex]
7. Check the condition that [tex]\(4\theta\)[/tex] must be acute:
[tex]\[
4\theta = 4 \times 78^\circ = 312^\circ
\][/tex]
This is not an acute angle (an angle less than 90°), which means there is no valid solution here that makes [tex]\(4\theta\)[/tex] an acute angle in this cycle.
8. Check for possible solutions using the negative cycle conversion (which does not yield valid acute angles either):
Therefore, given our criteria, there are no valid solutions where [tex]\(4\theta\)[/tex] is an acute angle matching the condition provided in the problem.
Thus, there is no value of [tex]\(\theta\)[/tex] that satisfies the equation [tex]\(\sec(4\theta) = \operatorname{cosec}(\theta - 300^\circ)\)[/tex] while keeping [tex]\(4\theta\)[/tex] as an acute angle. The answer is that no valid [tex]\( \theta \)[/tex] exists under these conditions.
