Let's find the value of [tex]\( k \)[/tex] given the equations [tex]\( \cos \theta = \sqrt{3k} \)[/tex] and [tex]\( \sec \theta = 2 \)[/tex].
We start with the relationship between cosine and secant:
[tex]\[
\text{sec}(\theta) = \frac{1}{\cos(\theta)}
\][/tex]
Since [tex]\( \sec \theta = 2 \)[/tex], we can write:
[tex]\[
2 = \frac{1}{\cos(\theta)}
\][/tex]
Solving for [tex]\(\cos(\theta)\)[/tex], we get:
[tex]\[
\cos(\theta) = \frac{1}{2}
\][/tex]
Now, we also have the equation:
[tex]\[
\cos(\theta) = \sqrt{3k}
\][/tex]
We can substitute [tex]\(\cos(\theta) = \frac{1}{2}\)[/tex] into the equation:
[tex]\[
\frac{1}{2} = \sqrt{3k}
\][/tex]
Next, we square both sides to eliminate the square root:
[tex]\[
\left(\frac{1}{2}\right)^2 = (\sqrt{3k})^2
\][/tex]
This simplifies to:
[tex]\[
\frac{1}{4} = 3k
\][/tex]
Finally, we solve for [tex]\( k \)[/tex] by dividing both sides by 3:
[tex]\[
k = \frac{1}{4 \times 3}
\][/tex]
[tex]\[
k = \frac{1}{12}
\][/tex]
Therefore, the value of [tex]\( k \)[/tex] is:
[tex]\[
k = 0.08333333333333333
\][/tex]
