SOLUTION: Find the exact value of the expression or state that it does not exist:
csc^-1 (-2)
a.The value of csc^-1 x is an angle in what interval?
b. Let csc^-1(-2) = θ such th
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Question 1099758: Find the exact value of the expression or state that it does not exist:
csc^-1 (-2)
a.The value of csc^-1 x is an angle in what interval?
b. Let csc^-1(-2) = θ such that sin^-1 = ___ and sinθ= ____
c. Is the value of sinθ positive, negative, or equal to 0?
d. does the angle θ: lie in the interval 0<θ<π/2 , is equal to π/2, or lies in the interval -π/2<θ<0
e. Where does the terminal side of angle θ lie?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
(alternatively represented as
is the inverse function of .
The graph of looks like this:
.
The function is a periodic function,
like all trigonometric functions,
so the inverse can only be defined by restricting the domains for the values, so as not to repeat values.
That would be taking only the part of the graph between the blue lines,
to get a function defined as
(quadrants I and IV).
Then we, can interchange the variables and define
is the angle ,
between and
such that
That is the same approach used to define the inverse of .
So,
a. The value of the function is an angle in the interval or .
b. What is sin^-1 supposed to mean? ?
is the angle , with that has
and .
NOTE:
We know that in quadrant I (or .
In quadrant IV, using the symmetrical quadrant I angle as reference,
we find that ,
so .
However, this question wants you to crawl to the answer very, very slowly.
c. Is the value of sinθ positive, negative, or equal to 0?
In part b. we found that we are looking for an angle whose sine is , so we know that is .
d. An angle that has a negative sine,
and is in the interval we agreed to in part b.
has to be in the interval
e. The terminal side of angle lies in quadrant IV.
Here it is:
It is a negative angle because it is a counterclockwise turn from the positive x-axis.
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