Understand inverse secant cosecant and cotangent functions

Question

Evaluate the following expression. arccsc(-1)

Answer 1

arccsc(-1) =about 3.141593/2 = - 1.57079 radians or

= -pi/2 radians or

= -90 degrees

arccscx = csc^-1(x)

= the inverse cosecant of x

= the angle whose cosecant = x

if you solve for x = csc(-1)

which also = sin(-1)

you'd get -pi/2 + 2npi where n= any integer

However, the solution to the arccsc(-1) does not include the +2npi term, because the domain of x is restricted to the interval [-pi/2, pi/2]

or -pi/2 < x < pi/2

you could graph the cosecant function and see where it has a value of -1. There are an infinity number of angles that do that, but you want only x= -pi/2

for arcsec(-1) = x

x=arccos(-1) = -pi = -180 degrees = about -3.141593 radians

domain for x is [-1,1] so don't add the 2npi or 360n or 2 terms

for arctan(-1)

that's the angle whose tangent =-1

which is in quadrants II and IV

-45 degrees + 180n where n= any integer

or - pi/4 + npi where n = any integer

or about - 3.141593/4 = -0.785398 + 3.141593 radians

BUT leave off the term with 180n, npi or 3.141593

as the domain is restricted for arctanx to the interval [-pi/2, pi/2]

-pi/2 < x < pi/2

so the answer is -45 degrees, -pi/4, or about -0.785398 radians

Answer 2

arc csc (-1)

arc sin (-1)

sin θ = -1

Can you determine sin θ and answer?