Simplify sec(arcsin((2 square root of 6)/9))

Problem

sec(arcsin((2√(,6))/9))

Solution

1. Identify the inner expression as an angle θ=arcsin((2√(,6))/9) which implies sin(θ)=(2√(,6))/9

2. Relate the sine function to the sides of a right triangle where sin(θ)=opposite/hypotenuse Let the opposite side be 2√(,6) and the hypotenuse be 9

3. Calculate the adjacent side a using the Pythagorean theorem a2+b2=c2

a2+(2√(,6))2=9

a2+24=81

a2=57

a=√(,57)

4. Determine the secant of the angle, where sec(θ)=hypotenuse/adjacent

sec(θ)=9/√(,57)

5. Rationalize the denominator by multiplying the numerator and denominator by √(,57)

9/√(,57)⋅√(,57)/√(,57)=(9√(,57))/57

6. Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 3

(3√(,57))/19

Final Answer

sec(arcsin((2√(,6))/9))=(3√(,57))/19

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