Final answer:
To find the exact value of csc a in simplest radical form using a rational denominator, we can use the Pythagorean Identity to solve for sin a and then apply the reciprocal identity csc a = 1/sin a. The exact value of csc a is √53/2.
Step-by-step explanation:
To find the exact value of csc a in simplest radical form using a rational denominator, we first need to determine the value of sin a. Since cos a = 7/√53, we can use the Pythagorean Identity sin^2 a + cos^2 a = 1 to solve for sin a. Rearranging the equation, we have sin^2 a = 1 - cos^2 a. Plugging in the given value of cos a, we can solve for sin a.
sin^2 a = 1 - (7/√53)^2
sin^2 a = 1 - 49/53
sin^2 a = (53 - 49)/53
sin^2 a = 4/53
Now, taking the square root of both sides to find sin a, we have sin a = √(4/53).
To find the value of csc a, we can use the reciprocal identity csc a = 1/sin a. Therefore, csc a = 1/(√(4/53)). To simplify this expression, we rationalize the denominator by multiplying the numerator and denominator by √53.
csc a = 1/(√(4/53)) * (√53/√53)
csc a = √53/√(4/53)
Using the property √(a/b) = √a/√b, we can simplify the expression further to csc a = √53/(√4/√53).
Since √4 = 2, we have csc a = √53/2.