Stitz-Zeager_College_Algebra_e-book

750 applications of trigonometry example 1112

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Unformatted text preview: if and only if − π ≤ t < 0 or 0 < t ≤ 2 – arccsc(x) = arcsin 1 x π 2 and csc(t) = x provided |x| ≥ 1 – csc (arccsc(x)) = x provided |x| ≥ 1 – arccsc(csc(x)) = x provided − π ≤ x < 0 or 0 < x ≤ 2 π 2 – additionally, arccosecant is odd a . . . assuming the “Trigonometry Friendly” ranges are used. Example 10.6.3. 1. Find the exact values of the following. 5π 4 (a) arcsec(2) (c) arcsec sec (b) arccsc(−2) (d) cot (arccsc (−3)) 2. Rewrite the following as algebraic expressions of x and state the domain on which the equivalence is valid. (a) tan(arcsec(x)) (b) cos(arccsc(4x)) Solution. 1. (a) Using Theorem 10.28, we have arcsec(2) = arccos 1 2 = π. 3 710 Foundations of Trigonometry (b) Once again, Theorem 10.28 comes to our aid giving arccsc(−2) = arcsin − 1 = − π . 2 6 π (c) Since 54 doesn’t fall between 0 and π or π and π , we cannot use the inverse property 2 2 stated in Theorem 10.28. We can, nevertheless, begin by working ‘inside out’ which √ √ π...
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