Which inverse trigonometric function has a range of [0,pi]?
a. y=cos^-1(x)
b. y=cot^-1(x)
c. y=csc^-1(x)
d. y=sin^-1(x)
I am not fully understanding this question. If someone could help me that would be great!
The inverse trigonometric functions are denoted by the prefix "arc" or with a superscript "-1". The range of an inverse trigonometric function represents the set of all possible values that the function can take.
To determine which inverse trigonometric function has a range of [0, pi], we need to analyze the given options:
a. y = cos^-1(x):
The range of the arccosine function, cos^-1(x), is [0, pi].
b. y = cot^-1(x):
The range of the arccotangent function, cot^-1(x), ranges from 0 to pi.
c. y = csc^-1(x):
The range of the arccosecant function, csc^-1(x), does not fall within the range [0, pi]. Therefore, this option is not correct.
d. y = sin^-1(x):
The range of the arcsine function, sin^-1(x), is [-pi/2, pi/2], not [0, pi]. Therefore, this option is also incorrect.
Based on the analysis, the correct answer is option a. y = cos^-1(x).
Sure! Let's break down the question and discuss each option.
The inverse trigonometric functions are used to find the angle based on a given trigonometric ratio. Each inverse trigonometric function has a specific range of values for which it is defined. The four commonly used inverse trigonometric functions are:
a. y = cos^(-1)(x) or arccos(x)
b. y = cot^(-1)(x) or arccot(x)
c. y = csc^(-1)(x) or arccsc(x)
d. y = sin^(-1)(x) or arcsin(x)
In this question, you are asked to find which inverse trigonometric function has a range of [0, π].
To determine the range, we need to consider the range of the corresponding trigonometric function and restrict it accordingly.
The range of the cosine function (cos(x)) is [-1, 1]. Therefore, the range of its inverse function, arccos(x), is [0, π]. So, option a. y = cos^(-1)(x) matches the given range [0, π]. It is the correct answer.
The other options, b. y = cot^(-1)(x), c. y = csc^(-1)(x), and d. y = sin^(-1)(x), do not have a range of [0, π]. Their ranges are different and not applicable to the given range.
So, the correct answer is a. y = cos^(-1)(x).
since cos(x) goes from cos(0)=1 to cos(pi) = -1 the domain of arccos(x) is [0,pi]
The others either get undefined or repeat their range values
the inverse of cosecant is sine , which is continuous from zero to π
the others are undefined at some point over the range