Stitz-Zeager_College_Algebra_e-book

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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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