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inversetrig - The Inverse Trigonometric Functions These...

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The Inverse Trigonometric Functions These notes amplify on the book’s treatment of inverse trigonometric functions and supply some needed practice problems. Please see pages 543– 544 for the graphs of sin - 1 x , cos - 1 x , and tan - 1 x . 1 The Arcsine Function My sine is x ; who am I? If x is any real number such that | x | ≤ 1, there are infinitely many possible answers. For example, let x = 1 / 2. Then sin y = x when y = π/ 6, 5 π/ 6, - 7 π/ 6, - 11 π/ 6, 13 π/ 6, 17 π/ 6, - 19 π/ 6, and so on. Note however that as y travels from - π/ 2 to π/ 2, sin y travels from - 1 to 1. Since sin y is continuous and increasing in the interval - π/ 2 y π/ 2, it follows that for any x between - 1 and 1 there is exactly one y between - π/ 2 and π/ 2 such that sin y = x . We can therefore define a new function sin - 1 as follows: Definition 1. If - 1 x 1, then sin - 1 x (also known as arcsin x ) is the number between - π/ 2 and π/ 2 whose sine is equal to x . Comment. If the function f has an inverse, that inverse is generally de- noted by f - 1 . The notation sin - 1 x that has just been introduced is (more or less) in accord with this general convention. Not quite! The sine func- tion does not have an inverse, since given sin t it is not possible to recover t uniquely. Maybe we are being too fussy. But the notation can also be a source of confusion. For note that sin 2 x is a standard abbreviation for (sin x ) 2 , and sin 3 x is a standard abbreviation for (sin x ) 3 . So should sin - 1 x mean (sin x ) - 1 ? Maybe it should. But it doesn’t! The notation arcsin x is preferred by many mathematicians. Unfortu- nately, sin - 1 seems to be gaining ground over arcsin, maybe because it fits the cramped space on calculator keyboards better. There are several other notations, including “Arc sin,” and “asin.” 1
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Pronunciations vary: sin - 1 x can be pronounced “sine inverse (of) x ,” “inverse sine (of) x ,” or even “arc sine x .” Comment. You don’t have to know a lot about geography to know the capital of the country whose capital is Amman. And you don’t have to know much about trigonometry to find sin(sin - 1 (0 . 123)). Indeed it is clear that sin(sin - 1 x ) = x for all x between -
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