2 3 \u03c0 1 3 2 \u03c0 Precalculus 64ASSIGNMENT 15 INVERSE TRIGONOMETRIC FUNCTIONS Read

2 3 π 1 3 2 π precalculus 64assignment 15 inverse

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Precalculus64ASSIGNMENT 15: INVERSETRIGONOMETRIC FUNCTIONSRead the following information. Then study pages 550–562 inyour textbook, Precalculus.Complete the “Check Points” as youcome to them in the textbook. The answers are in the back ofyour textbook.IntroductionHere are some reminders that will be of use to you as youbegin this section on inverse trigonometric functions.If the name of a function is f, then its inverse is f –1. Note: The superscript is notan exponent in this context.If a point (a, b) is on the graph of a function, then thepoint (b, a) is on the graph of its inverse. The two graphswill be reflections of each other about the diagonal line y = x.Only one-to-one functions have inverse functions. Inother words, if a horizontal line intersects the graph of a function in more than one point, that function won’t have an inverse.Self-Check 14To reinforce what you’ve just studied, complete exercises 1, 5, 9, 13, 17, 21, 23, 25, 27, 29,33, 37, 39, 41, and 43 on pages 546–547 of your textbook. Check your answers with those in the back of your textbook. If you make a mistake, analyzeyour work and review the related material in this study guide and in your textbook.
Lesson 265The Inverse Sine FunctionThe function y= sin xisn’t a one-to-one function. If you weregiven a y-value on the graph of y= sin x,there would be infi-nitely many x-values that correspond to it. This principle isillustrated in the horizontal line test(Figure 4.86 on page 550).So the reflection of y= sin xabout the line y = xwould notdefine a function.However, suppose you take only a portion of the sine wave—say, . This portion of the sine wave does represent a one-to-one function. To refer to the inverse of this restrictedpart of the graph of y= sin x,use the notation y= sin–1x.Think of xas a numberbetween –1 and 1 and yas the anglebetween and whose sine is x.So y= sin–1xmeans sin y = x. The function y= sin–1x is read “yequals theinverse sine at x.” (In other textbooks and on some calculators,you might see sin–1 written as arcsin.) The blue box on page535 states this in a concise manner.To evaluate the inverse sine function, for example, , think of an angle whose sine is. In other words, you haveto think backwards. The angle is . The Inverse Cosine and Tangent FunctionsNow that you’ve studied the rules for finding the inverse sineof a number, you can apply them to both the cosine and thetangent functions. The ideas remain the same. However,notice that the cosine function (see page 553) isn’t one-to-onein the interval . Therefore, to take a one-to-one portionof the graph of y= cos x,you use the interval [0, ] instead.When you take the inverse cosine of a number (an angle), theanswer will be restricted to the interval [0, ]. The exactvalues of the inverse cosine function are given on page 554.

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