Simplify cot(arcsin(-( square root of 2)/3))

Problem

cot(arcsin(−√(,2)/3))

Solution

1. Identify the inner expression as an angle θ=arcsin(−√(,2)/3) By the definition of the inverse sine function, sin(θ)=−√(,2)/3 where θ is in the interval [−π/2,π/2]

2. Determine the quadrant of θ Since the sine value is negative, θ must be in the fourth quadrant (Quadrant IV), where cos(θ) is positive and cot(θ) is negative.

3. Use the Pythagorean identity sin2(θ)+cos2(θ)=1 to find cos(θ)

(−√(,2)/3)2+cos2(θ)=1

2/9+cos2(θ)=1

cos2(θ)=1−2/9

cos2(θ)=7/9

4. Solve for cos(θ) Since θ is in Quadrant IV, cos(θ) is positive.

cos(θ)=√(,7/9)

cos(θ)=√(,7)/3

5. Apply the definition of the cotangent function, cot(θ)=cos(θ)/sin(θ)

cot(θ)=√(,7)/3/(−√(,2)/3)

6. Simplify the fraction by canceling the denominators.

cot(θ)=−√(,7)/√(,2)

7. Rationalize the denominator by multiplying the numerator and denominator by √(,2)

cot(θ)=−(√(,7)⋅√(,2))/(√(,2)⋅√(,2))

cot(θ)=−√(,14)/2

Final Answer

cot(arcsin(−√(,2)/3))=−√(,14)/2

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