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**Unformatted text preview: **) = x for all real numbers x
– arccot(cot(x)) = x provided 0 < x < π
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 and
2
√
t = π , so arctan( 3) = π .
3
3 π
2 with tan(t) = √ 3. We ﬁnd 10.6 The Inverse Trigonometric Functions 707 √
√
(b) The real √
number t = arccot(− 3) lies in the interval (0, π ) with cot(t) = − 3. We get
π
arccot(− 3) = 56 .
(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), we have that t belongs to the interval (0, π ) and
cot(t) = −5. In terms of t, the expression cot(arccot(−5)) = cot(t), and since cot(...

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