Since arcsin x is defined only for 1 x 1 the

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Chapter 3 / Exercise 17
Single Variable Calculus: Early Transcendentals
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Since arcsin( x ) is defined only for - 1 x 1, the equivalence cos (2 arcsin( x )) = 1 - 2 x 2 is valid only on [ - 1 , 1]. A few remarks about Example 10.6.1 are in order. Most of the common errors encountered in dealing with the inverse circular functions come from the need to restrict the domains of the original functions so that they are one-to-one. One instance of this phenomenon is the fact that arccos ( cos ( 11 π 6 )) = π 6 as opposed to 11 π 6 . This is the exact same phenomenon discussed in Section 5.2 when we saw p ( - 2) 2 = 2 as opposed to - 2. Additionally, even though the expression we arrived at in part 2b above, namely 1 - 2 x 2 , is defined for all real numbers, the equivalence cos (2 arcsin( x )) = 1 - 2 x 2 is valid for only - 1 x 1. This is akin to the fact that while the expression x is defined for all real numbers, the equivalence ( x ) 2 = x is valid only for x 0. For this reason, it pays to be careful when we determine the intervals where such equivalences are valid. The next pair of functions we wish to discuss are the inverses of tangent and cotangent, which are named arctangent and arccotangent, respectively. First, we restrict f ( x ) = tan( x ) to its fundamental cycle on ( - π 2 , π 2 ) to obtain f - 1 ( x ) = arctan( x ). Among other things, note that the vertical asymptotes x = - π 2 and x = π 2 of the graph of f ( x ) = tan( x ) become the horizontal asymptotes y = - π 2 and y = π 2 of the graph of f - 1 ( x ) = arctan( x ). x y - π 4 - π 2 π 4 π 2 - 1 1 f ( x ) = tan( x ), - π 2 < x < π 2 . reflect across y = x ------------→ switch x and y coordinates x y - π 4 - π 2 π 4 π 2 - 1 1 f - 1 ( x ) = arctan( x ). Next, we restrict g ( x ) = cot( x ) to its fundamental cycle on (0 , π ) to obtain g - 1 ( x ) = arccot( x ). Once again, the vertical asymptotes x = 0 and x = π of the graph of g ( x ) = cot( x ) become the horizontal asymptotes y = 0 and y = π of the graph of g - 1 ( x ) = arccot( x ). We show these graphs on the next page and list some of the basic properties of the arctangent and arccotangent functions.
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Chapter 3 / Exercise 17
Single Variable Calculus: Early Transcendentals
Stewart Expert Verified
824 Foundations of Trigonometry x y π 4 π 2 3 π 4 π - 1 1 g ( x ) = cot( x ), 0 < x < π . reflect across y = x ------------→ switch x and y coordinates x y π 4 π 2 3 π 4 π - 1 1 g - 1 ( x ) = arccot( x ). Theorem 10.27. Properties of the Arctangent and Arccotangent Functions Properties of F ( x ) = arctan( x ) Domain: ( -∞ , ) Range: ( - π 2 , π 2 ) as x → -∞ , arctan( x ) → - π 2 + ; as x → ∞ , arctan( x ) π 2 - arctan( x ) = t if and only if - π 2 < t < π 2 and tan( t ) = x arctan( x ) = arccot ( 1 x ) for x > 0 tan (arctan( x )) = x for all real numbers x arctan(tan( x )) = x provided - π 2 < x < π 2 additionally, arctangent is odd Properties of G ( x ) = arccot( x ) Domain: ( -∞ , ) Range: (0 , π ) as x → -∞ , arccot( x ) π - ; as x → ∞ , arccot( x ) 0 + arccot( x ) = t if and only if 0 < t < π and cot( t ) = x arccot( x ) = arctan ( 1 x ) for x > 0 cot (arccot( x )) = x for all real numbers x arccot(cot( x )) = x provided 0 < x < π
10.6 The Inverse Trigonometric Functions 825 Example 10.6.2.
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