College Trigonometry, 2011a

820 Foundations of Trigonometry
We restrict g(x) =sin(x) in a similar manner, although the interval of choice is [ − π 2 , π ]
2 .
y
x
Restricting the domain of f(x) =sin(x) to [ − π 2 , π 2
]
.
It should be no surprise that we call g −1 (x) = arcsin(x),whichisread‘arc-sineofx’.
y
π
2
y
1
− π 2
π
2
x
−1 1
x
−1
g(x) = sin(x), − π 2 ≤ x ≤ π 2 .
reflect across y = x
−−−−−−−−−−−−→
switch x and y coordinates
− π 2
g −1 (x) = arcsin(x).
We list some important facts about the arccosine and arcsine functions in the following theorem.
Theorem 10.26. Properties of the Arccosine and Arcsine Functions
ˆ Properties of F (x) = arccos(x)
– Domain: [−1, 1]
– Range: [0,π]
– arccos(x) =t ifandonlyif0≤ t ≤ π and cos(t) =x
– cos(arccos(x)) = x provided −1 ≤ x ≤ 1
– arccos(cos(x)) = x provided 0 ≤ x ≤ π
ˆ Properties of G(x) = arcsin(x)
– Domain: [−1, 1]
– Range: [ − π 2 , π ]
2
– arcsin(x) =t if and only if − π 2 ≤ t ≤ π 2
and sin(t) =x
– sin(arcsin(x)) = x provided −1 ≤ x ≤ 1
– arcsin(sin(x)) = x provided − π 2 ≤ x ≤ π 2
– additionally, arcsine is odd