To determine whether [tex]\(\frac{3 \pi}{4}\)[/tex] is a solution for the equation [tex]\(3 \sqrt{2} \sec \theta + 7 = 1\)[/tex], we need to follow these steps:
1. Substitute [tex]\(\theta = \frac{3 \pi}{4}\)[/tex] into the equation.
2. Calculate [tex]\(\sec \left( \frac{3 \pi}{4} \right)\)[/tex].
3. Plug this value into the equation to verify if the equality holds true.
Step 1: Substitute [tex]\(\theta = \frac{3 \pi}{4}\)[/tex]
Start by considering the angle [tex]\(\theta = \frac{3 \pi}{4}\)[/tex].
Step 2: Calculate [tex]\(\sec \left( \frac{3 \pi}{4} \right)\)[/tex]
Recall that [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex].
For [tex]\(\theta = \frac{3 \pi}{4}\)[/tex], we have:
[tex]\[
\cos \left( \frac{3 \pi}{4} \right) = -\frac{\sqrt{2}}{2}
\][/tex]
Thus:
[tex]\[
\sec \left( \frac{3 \pi}{4} \right) = \frac{1}{\cos \left( \frac{3 \pi}{4} \right)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2}
\][/tex]
Step 3: Plug [tex]\(\sec \left( \frac{3 \pi}{4} \right) = -\sqrt{2}\)[/tex] into the equation
We now substitute [tex]\(\sec \theta = -\sqrt{2}\)[/tex] back into the original equation:
[tex]\[
3 \sqrt{2} \sec \left( \frac{3 \pi}{4} \right) + 7 = 3 \sqrt{2} (-\sqrt{2}) + 7
\][/tex]
Simplify the expression:
[tex]\[
3 \sqrt{2} \cdot (-\sqrt{2}) + 7 = 3 \cdot (-2) + 7 = -6 + 7 = 1
\][/tex]
We see that the equation holds true:
[tex]\[
1 = 1
\][/tex]
Since this equality is satisfied, we conclude that [tex]\(\frac{3 \pi}{4}\)[/tex] is indeed a solution to the given equation. Therefore, the answer is:
A. True
