calculus expressions

Question

_{Evaluate the expression without using a calculator.}

 

arccsc (-√2)

Answer 1

Let

 

x = csc^-1(-√2)

 

Then

 

csc(x) = -√2

 

1/sin(x) = -√2

 

sin(x) = -1/√2 = -√2 / 2

 

Recall sin(π/4) = √2 / 2

 

Then

 

x = -π/4 ± 2πn, with n = integer