Answer:
[tex]\csc A = \dfrac{\sqrt{41}}{5}[/tex]
Step-by-step explanation:
To find the exact value of csc A in simplest radical form using a rational denominator, we can use the reciprocal identities of secant and cosecant:
[tex]\boxed{\sec A = \dfrac{1}{\cos A}}\;\;\;\boxed{\text{csc} A = \dfrac{1}{\sin A}}[/tex]
Given sec A = √(41)/4, we can use the definition of secant as the reciprocal of cosine to find the value of cos A:
[tex]\cos A=\dfrac{1}{\sec A}=\dfrac{1}{\frac{\sqrt{41}}{4}}=\dfrac{4}{\sqrt{41}}[/tex]
Since angle A is in Quadrant I, both sin A and cos A are positive.
Now we can substitute the value of cos A into the trigonometric identity to find sin A:
[tex]\sin^2 A + \cos^2 A = 1[/tex]
[tex]\sin^2 A + \left(\dfrac{4}{\sqrt{41}}\right)^2 = 1[/tex]
[tex]\sin^2 A + \dfrac{16}{41} = 1[/tex]
[tex]\sin^2 A = 1-\dfrac{16}{41}[/tex]
[tex]\sin^2 A =\dfrac{25}{41}[/tex]
[tex]\sin A=\sqrt{\dfrac{25}{41}}[/tex]
[tex]\sin A=\dfrac{5}{\sqrt{41}}[/tex]
Finally, we can substitute the value of sin A into the reciprocal identity to find csc A:
[tex]\text{csc} A = \dfrac{1}{\sin A} = \dfrac{1}{\frac{5}{\sqrt{41}}} = \dfrac{\sqrt{41}}{5}[/tex]
Therefore, the exact value of csc A in simplest radical form with a rational denominator is:
[tex]\boxed{\csc A = \dfrac{\sqrt{41}}{5}}[/tex]
