To solve the equation [tex]\( 5 \cos \theta = -2 \)[/tex] in the range [tex]\( 180^\circ \leq \theta \leq 360^\circ \)[/tex], follow the detailed steps below:
1. Isolate [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos \theta = \frac{-2}{5} \][/tex]
[tex]\[ \cos \theta = -0.4 \][/tex]
2. Determine the reference angle:
Calculate the angle for [tex]\(\theta\)[/tex] where [tex]\(\cos \theta = -0.4\)[/tex] within the standard range of [tex]\(0^\circ \leq \theta < 360^\circ\)[/tex].
The angle [tex]\(\theta\)[/tex] given by [tex]\(\cos^{-1} (-0.4)\)[/tex] traditionally falls within the range [tex]\(0^\circ \leq \theta \leq 180^\circ\)[/tex], and it is approximately:
[tex]\[ \theta_1 \approx 113.578^\circ \][/tex]
3. Determine the angles in the specified range:
Since [tex]\(\cos \theta\)[/tex] is negative in the second quadrant (between [tex]\(90^\circ\)[/tex] and [tex]\(180^\circ\)[/tex]) and in the third quadrant (between [tex]\(180^\circ\)[/tex] and [tex]\(270^\circ\)[/tex]), we need to find the corresponding angles within the range [tex]\(180^\circ \leq \theta \leq 360^\circ\)[/tex].
- The first angle in the specified range can be found by considering the angle [tex]\(\theta\)[/tex] from the third quadrant that corresponds to the reference angle found above.
[tex]\[ \theta_{2} = 180^\circ + \theta_1 \approx 180^\circ + 113.578^\circ \approx 293.578^\circ \][/tex]
- The second angle is found by symmetry in the range [tex]\(360^\circ - \theta_1\)[/tex]:
[tex]\[ \theta_{3} = 360^\circ - \theta_1 \approx 360^\circ - 113.578^\circ \approx 246.422^\circ \][/tex]
4. Final angles:
Therefore, the solutions for [tex]\( \theta \)[/tex] in the range [tex]\( 180^\circ \leq \theta \leq 360^\circ \)[/tex] such that [tex]\( 5 \cos \theta = -2 \)[/tex] are:
[tex]\[
\theta = 293.578^\circ \quad \text{and} \quad \theta = 246.422^\circ
\][/tex]
These angles satisfy the given equation in the specified range.
