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# Class Note for MATH 1330 at UH 2

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Date Created: 02/06/15

Math 1330 Section 54 Inverse Trigonometric Functions We are often interested in nding and working with the inverses of functions This will also be the case with trig functions But there s a problem For a function to have an inverse it must be onetoone and trig functions are not onetoone To get around this we begin by limiting the domain of each of the trig functions so that over the restricted domain the function is onetoone We ll work with the inverse functions for sine cosine and tangent in the streaming lectures The reciprocal functions are treated in the online text We ll start by looking at the inverse sine function Here are the graphs of the functions f x sinx and gx g on the interval 271 271 This confirms that f x sinx is not a onetoone function range ofthis function is l 1 If we limit the domain of the function to 2 7Z7 then the functlon 1s onetoone The So on this limited interval we can define an inverse function We note the inverse sine function as So on this limited interval we can define an inverse function We note the inverse sine function as f x sin 1 x or f x arcsinx The domain of the inverse function will be 1 l and the range of the inverse function will be Here is the graph of fx sin 1 x Next we ll look at the inverse cosine function Since the cosine function is also not one toone we ll need to restrict the domain to an interval which will give us a onetoone function Then it makes sense to talk about an inverse function We can t use the same interval that we did for the inverse sine function however This interval does not produce a onetoone function Instead we ll use the interval 0 7 Here s the graph of f x cosx on the interval 0 It The range of this function is 41 74 So on this limited interval we can define an inverse function We note the inverse cosine function as fx cos 1 x or fx arccosx The domain of the inverse function Will be 1 l and the range of the inverse function will be 7r Here is the graph of fx cos 1 x Finally we ll look at the graph of the tangent function We ll restrict the domain to the 77quot 77quot interval Ejso that we can de ne an inverse function Here is the graph of fx tanx on the interval The range of this function is 0 00 J We can find an inverse tangent function on this limited domain We note the function as fx tan 1 x or fx arctanx The domain of the inverse function is 0 0 and 77quot 77quot the range is Here is the graph of the inverse tangent function Most o en you will be expected to evaluate inverse trig functions That is given a problem such as sin391 x y you will want to nd the number in the interval 2 2 sin y x and then use the chart we developed for working with unit circle values You will need to note the domain when working with problems ofthis type In some cases you can draw and label a right triangle in the appropriate quadrant or you can work with identities to evaluate These properties may be helpful when evaluating inverse trig problems 1 1 whose sine is x The simplest way to do this will be to rewrite the problem as sinsin39lx x on 71 l coscos39l x x on 71 l tantan39l 0 x on 7 cc 00 sinquot sinx x on cosquot cosx x on 0 7r 7 5 tanquot tanx x on 2 Finally you will sometimes need to use a calculator Note that you will always use radians mode when evaluating inverse trig functions on your calculator Example 1 Evaluate cos39l Example 2 Evaluate arctan l Example 3 Evaluate sin 1T Example 4 Evaluate sec 1 2 Example 5 Evaluate csc 10 Example 6 Find the exact value sincos 1 Example 7 Find the exact value cos 1cos3 j Example 8 Find the exact value tansin 174 Example 9 Find the exact value cotsec 1 Example 10 Use a calculator to nd the exact value and round the answer to the nearest thousandth arcsin75 Example 11 Use a calculator to find the exact value and round the answer to the nearest thousandth cos 1 7 l 8 Example 12 Use a calculator to find the exact value and round the answer to the nearest thousandth sec 1 Example 13 Use a calculator to find the exact value and round the answer to the nearest thousandth cot 1 14 You can use graphing techniques learned in earlier lessons to graph transformations of the basic inverse trig functions Example 14 Sketch fx cos 1x 2

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