To solve the equation [tex]$\cos \Theta = -0.25$[/tex] with the restriction [tex]$180^{\circ} < \Theta < 270^{\circ}$[/tex], we will follow these steps:
1. Inverse Cosine Calculation:
First, we need to find the angle whose cosine is [tex]\(-0.25\)[/tex]. Using the inverse cosine function (also known as arccos), we find the principal value:
[tex]\[
\Theta_1 = \cos^{-1}(-0.25)
\][/tex]
This principal value will be an angle in the first or fourth quadrant.
2. Convert Radians to Degrees:
In most cases, the output from the arccos function will be in radians, so it's essential to convert this angle to degrees. Suppose the principal value of [tex]\(\cos^{-1}(-0.25)\)[/tex] converts to approximately 104.5°.
3. Identify Relevant Quadrant:
The principal value of 104.5° is in the second quadrant. Since the cosine function is symmetric, we also get an angle in the third quadrant given by:
[tex]\[
\Theta_2 = 360^{\circ} - 104.5^{\circ} = 255.5^{\circ}
\][/tex]
4. Check the Restriction:
We need the angle to be within the restriction [tex]\(180^{\circ} < \Theta < 270^{\circ}\)[/tex]. The angle 255.5° falls within this range, thus it qualifies as our solution.
5. Conclusion:
Therefore, the solution for [tex]\(\Theta\)[/tex] within the restriction [tex]\(180^{\circ} < \Theta < 270^{\circ}\)[/tex], where [tex]\(\cos \Theta = -0.25\)[/tex], rounded to the nearest tenth, is:
[tex]\[
\boxed{255.5^{\circ}}
\][/tex]
