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Given sin A = sqr rt 41/ 4 and that angle A is in Quadrant I, find the exact value of sec⁡ A in simplest radical form using a rational denominator.

Given sin A = sqr rt 41/ 4 and that angle A is in Quadrant I, find the exact value-example-1

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Answer:


\csc A = (√(41))/(5)

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:


\boxed{\sec A = (1)/(\cos A)}\;\;\;\boxed{\text{csc} A = (1)/(\sin A)}

Given sec A = √(41)/4, we can use the definition of secant as the reciprocal of cosine to find the value of cos A:


\cos A=(1)/(\sec A)=(1)/((√(41))/(4))=(4)/(√(41))

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:


\sin^2 A + \cos^2 A = 1


\sin^2 A + \left((4)/(√(41))\right)^2 = 1


\sin^2 A + (16)/(41) = 1


\sin^2 A = 1-(16)/(41)


\sin^2 A =(25)/(41)


\sin A=\sqrt{(25)/(41)}


\sin A=(5)/(√(41))

Finally, we can substitute the value of sin A into the reciprocal identity to find csc A:


\text{csc} A = (1)/(\sin A) = (1)/((5)/(√(41))) = (√(41))/(5)

Therefore, the exact value of csc A in simplest radical form with a rational denominator is:


\boxed{\csc A = (√(41))/(5)}

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