cos (arccsc 1)?

Question

evaluate (to the exact value - no decimals)

please help me. thank you!

Answer 1

csc (x) = 1 when sin (x) = 1 which is at pi/2, and cos (pi/2) = 0

Answer 2

First some information.

csc(π/2) = 1
arccsc(1) = π/2

Now we can evaluate the expression.

cos(arccsc 1) = cos(π/2) = 0

Answer 3

cos (arccsc(1))

To solve this,

let t = arccsc(1). Then, take the csc of both sides.

csc(t) = csc[arccsc(1)]

Note that csc and arccsc are functional inverses of each other, and cancel each other out.

csc(t) = 1

Here's where we apply what we know about SOHCAHTOA.

sin = opp/hyp, cos = adj/hyp, tan = opp/adj

And by extension, since csc is the reciprocal of sin,

csc = hyp/opp

Therefore, think of csc(t) = 1 as
csc(t) = 1/1 = hyp/opp

Also, think of working at this problem as utilizing a right triangle. As per above,

hyp = 1
opp = 1, so by Pythagoras,
adj = sqrt(hyp^2 - opp^2) = sqrt(1 - 1) = 0

Therefore,
cos(t) = cos(arccsc(1)) = adj/hyp = 0/1 = 0

Answer 4

arccsc 1=arcsin 1/1=π/2
cos π/2=0