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Calculus Section 5.6 Inverse Trig Functions
-Develop properties of the six inverse trigonometric functions
Homework: page 377 #’s 5 – 27 odd, 31, 32, 33
(Hint: #33 take the sin of both sides first)
Fact: None of the six basic trigonometric functions (sin, cos, …) has an inverse functions. This is because all trig
functions are ___________________ and thus not _________________________.
However, you can ___________________ the domain for the trig functions to allow them to have an inverse.
For example, the sine function is one-to-one if its domain is restricted to __________________. Then, then
inverse function of sine is defined as:
The Six Trig and Inverse Trig Functions
sinx
cosx
arcsinx
arccosx
arctanx
cscx
secx
cotx
arccscx
arcsecx
arccotx
tanx
The inverse function arcsin can also be written as:
By definition, y = arcsin(-1/2) implies:
Properties of Inverse Trig Functions
If each trig function is restricted to its one-to-one domain, the following properties are true:
If -1 ≤ x ≤ 1 and –π/2 ≤ y ≤ π/2, then
If –π/2 < y < π/2, then
If -1 ≤ x ≤ 1 and 0 ≤ y < π/2 or π/2 < y ≤ π, then
Similar properties hold for the other inverse functions on their restricted domains.
Examples
1) arctan(2x – 3) = π/4
4) Find cos[arcsin(x)]
2) arcos(0) =
5) Given y = arcsec( 
3) sin-1(x2) = π/2
5
), find tan(y).
2