College Trigonometry, 2011a

10.6 The Inverse Trigonometric Functions 825
Example 10.6.2.
1. Find the exact values of the following.
(a) arctan( √ 3) (b) arccot(− √ 3)
(c) cot(arccot(−5)) (d) sin ( arctan ( − 3 ))
4
2. Rewrite the following as algebraic expressions of x and state the domain on which the equivalence
is valid.
(a) tan(2 arctan(x))
(b) cos(arccot(2x))
Solution.
1. (a) We know arctan( √ 3) is the real number t between − π 2 and π 2 with tan(t) =√ 3. We find
t = π 3 ,soarctan(√ 3) = π 3 .
(b) The real number t = arccot(− √ 3) lies in the interval (0,π) with cot(t) =− √ 3. We get
arccot(− √ 3) = 5π 6 .
(c) We can apply Theorem 10.27 directly and obtain cot(arccot(−5)) = −5. However,
working it through provides us with yet another opportunity to understand why this
is the case. Letting t = arccot(−5),wehavethatt belongs to the interval (0,π)and
cot(t) =−5. Hence, cot(arccot(−5)) = cot(t) =−5.
(d) We start simplifying sin ( arctan ( − 3 ( )
4))
by letting t =arctan −
3
4 . Then tan(t) =−
3
4 for
some − π 2