Simplify cot(arcsin(-( square root of 7)/8))

Problem

cot(arcsin(−√(,7)/8))

Solution

1. Identify the inner expression as an angle θ=arcsin(−√(,7)/8)

2. Determine the range of the inverse sine function, which is [−π/2,π/2] Since the argument −√(,7)/8 is negative, θ must be in the interval (−π/2,0) which corresponds to the fourth quadrant.

3. Use the definition of the sine function sin(θ)=y/r=−√(,7)/8 We can let y=−√(,7) and r=8

4. Calculate the adjacent side x using the Pythagorean theorem x2+y2=r2

x2+(−√(,7))2=8

x2+7=64

x2=57

5. Select the positive root x=√(,57) because the cosine (and thus the xcoordinate) is positive in the fourth quadrant.

6. Apply the definition of the cotangent function cot(θ)=x/y

cot(θ)=√(,57)/(−√(,7))

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

cot(θ)=−(√(,57)⋅√(,7))/7

cot(θ)=−√(,399)/7

Final Answer

cot(arcsin(−√(,7)/8))=−√(,399)/7

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