Let's work through the given problem step-by-step.
1. Given Information:
- [tex]\(\cos(x) = \frac{5}{13}\)[/tex]
- The angle [tex]\(x\)[/tex] is in the range [tex]\(270^\circ \leq x \leq 360^\circ\)[/tex]. This indicates that [tex]\(x\)[/tex] is in the 4th quadrant.
2. Characteristics in the 4th Quadrant:
- In the 4th quadrant, the cosine value is positive, which matches our given [tex]\(\cos(x) = \frac{5}{13}\)[/tex].
- The sine value in the 4th quadrant is negative.
3. Finding [tex]\(\sin(x)\)[/tex] Using the Pythagorean Identity:
- We use the Pythagorean identity: [tex]\(\cos^2(x) + \sin^2(x) = 1\)[/tex].
- [tex]\(\cos^2(x) = \left(\frac{5}{13}\right)^2 = \frac{25}{169}\)[/tex].
- Let [tex]\(\sin(x) = y\)[/tex]. Thus, we have:
[tex]\[
\left(\frac{5}{13}\right)^2 + y^2 = 1
\][/tex]
[tex]\[
\frac{25}{169} + y^2 = 1
\][/tex]
[tex]\[
y^2 = 1 - \frac{25}{169}
\][/tex]
[tex]\[
y^2 = \frac{169}{169} - \frac{25}{169}
\][/tex]
[tex]\[
y^2 = \frac{144}{169}
\][/tex]
- Taking the square root of both sides, we get:
[tex]\[
y = \pm\sqrt{\frac{144}{169}}
\][/tex]
[tex]\[
y = \pm\frac{12}{13}
\][/tex]
- Since [tex]\(x\)[/tex] is in the 4th quadrant where sine is negative, we have:
[tex]\[
y = -\frac{12}{13}
\][/tex]
4. Calculation for [tex]\(y + 6\)[/tex]:
- Now, we need to find [tex]\(y + 6\)[/tex]:
[tex]\[
y + 6 = -\frac{12}{13} + 6
\][/tex]
- Converting 6 to a fraction with the same denominator:
[tex]\[
y + 6 = -\frac{12}{13} + \frac{78}{13}
\][/tex]
- Combining the fractions:
[tex]\[
y + 6 = \frac{-12 + 78}{13}
\][/tex]
[tex]\[
y + 6 = \frac{66}{13}
\][/tex]
[tex]\[
y + 6 \approx 5.076923076923077
\][/tex]
So, the detailed solution yields the values:
- [tex]\(\cos(x) = \frac{5}{13} \approx 0.38461538461538464\)[/tex]
- [tex]\(\sin(x) = -\frac{12}{13} \approx -0.9230769230769231\)[/tex]
- The value of [tex]\(y + 6\)[/tex] is approximately [tex]\(5.076923076923077\)[/tex].
