Given the information that [tex]\(\csc A = -3\)[/tex] and [tex]\(\cos A > 0\)[/tex], we can find [tex]\(\sec A\)[/tex] by following these detailed steps:
1. Determine [tex]\(\sin A\)[/tex]:
[tex]\[\csc A = \frac{1}{\sin A}\][/tex]
Given [tex]\(\csc A = -3\)[/tex], we have:
[tex]\[ -3 = \frac{1}{\sin A} \][/tex]
Solving for [tex]\(\sin A\)[/tex], we get:
[tex]\[\sin A = -\frac{1}{3}\][/tex]
2. Use the Pythagorean identity:
The Pythagorean identity states:
[tex]\[\sin^2 A + \cos^2 A = 1\][/tex]
Substituting [tex]\(\sin A = -\frac{1}{3}\)[/tex], we find:
[tex]\[\left(-\frac{1}{3}\right)^2 + \cos^2 A = 1\][/tex]
[tex]\[\frac{1}{9} + \cos^2 A = 1\][/tex]
Solving for [tex]\(\cos^2 A\)[/tex], we have:
[tex]\[\cos^2 A = 1 - \frac{1}{9}\][/tex]
[tex]\[\cos^2 A = \frac{9}{9} - \frac{1}{9}\][/tex]
[tex]\[\cos^2 A = \frac{8}{9}\][/tex]
3. Solve for [tex]\(\cos A\)[/tex]:
Since [tex]\(\cos^2 A = \frac{8}{9}\)[/tex], we take the square root to find [tex]\(\cos A\)[/tex]. Given that [tex]\(\cos A > 0\)[/tex], we take the positive root:
[tex]\[\cos A = \sqrt{\frac{8}{9}}\][/tex]
[tex]\[\cos A = \frac{\sqrt{8}}{3}\][/tex]
Simplifying further:
[tex]\[\cos A = \frac{2\sqrt{2}}{3}\][/tex]
4. Determine [tex]\(\sec A\)[/tex]:
By definition:
[tex]\[\sec A = \frac{1}{\cos A}\][/tex]
Substituting the value of [tex]\(\cos A\)[/tex]:
[tex]\[\sec A = \frac{1}{\frac{2\sqrt{2}}{3}}\][/tex]
Simplifying the fraction:
[tex]\[\sec A = \frac{3}{2\sqrt{2}}\][/tex]
Rationalizing the denominator:
[tex]\[\sec A = \frac{3}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}\][/tex]
[tex]\[\sec A = \frac{3\sqrt{2}}{2 \cdot 2}\][/tex]
[tex]\[\sec A = \frac{3\sqrt{2}}{4}\][/tex]
Thus, the value of [tex]\(\sec A\)[/tex] is [tex]\(\frac{3\sqrt{2}}{2}\)[/tex], which numerically evaluates to approximately [tex]\(1.0606601717798212\)[/tex].
