To find [tex]\(\sec(\theta)\)[/tex] given that [tex]\(\tan(\theta) = 2\)[/tex] and [tex]\(\theta\)[/tex] lies in Quadrant II, we can use a Pythagorean identity and follow a detailed step-by-step approach.
1. Use the Pythagorean identity: The identity relating [tex]\(\tan(\theta)\)[/tex] and [tex]\(\sec(\theta)\)[/tex] is:
[tex]\[
1 + \tan^2(\theta) = \sec^2(\theta)
\][/tex]
2. Substitute the given value: We are given that [tex]\(\tan(\theta) = 2\)[/tex]. Substitute this value into the identity:
[tex]\[
1 + (2)^2 = \sec^2(\theta)
\][/tex]
3. Simplify:
[tex]\[
1 + 4 = \sec^2(\theta)
\][/tex]
[tex]\[
\sec^2(\theta) = 5
\][/tex]
4. Solve for [tex]\(\sec(\theta)\)[/tex]: To find [tex]\(\sec(\theta)\)[/tex], take the square root of both sides:
[tex]\[
\sec(\theta) = \pm\sqrt{5}
\][/tex]
5. Determine the sign based on the quadrant: Since [tex]\(\theta\)[/tex] is in Quadrant II, we know that [tex]\(\cos(\theta)\)[/tex] is negative (since cosine is negative in the second quadrant). Consequently, [tex]\(\sec(\theta)\)[/tex], which is the reciprocal of [tex]\(\cos(\theta)\)[/tex], will also be negative.
Putting it all together, we have:
[tex]\[
\sec(\theta) = -\sqrt{5}
\][/tex]
Therefore, the exact, fully simplified answer is:
[tex]\[
\sec(\theta) = -\sqrt{5}
\][/tex]
