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4.1.30001 - Definition The inverse Cosine Function 4.1 The...

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Unformatted text preview: Definition: The inverse Cosine Function 4.1 The Inverse Trigonometric Functions 213 The Inverse‘Cosine Function Since f (x) = cos(x) is not one-to—one on (—00, 00), we restrict the domain to [0, 77] where it is one—to-one and invertible. The graph of f (x) = cos(x) with this restricted domain is shown in Fig. 4.3(a). Note that the range of the restricted function is [—l, l]. The inverse off(x) = cos x forx in [0, ’77] is denoted as f_1(x) = cos_1(x) or f ”1(x) = arccos(x). If y = cos71(x), then y is the real number in [0, 11'] such that cos(y) = x. The domain of y = cos-1x is [—l, l] and its range is [0, 77]. The expression cos _1x can be read as “the angle whose cosine is x” or “the arc length whose cosine is x.” The graph of y = cos_1 x shown in Fig. 4.3(b) is obtained by reflecting the graph of y = cos x (restricted to [0, 77]) about the line y = x. We will assume that cos—1x is a real number unless indicated otherwise. Figure 4.3 EXAMPLE 4 Evaluating the inverse cosine function Find the exact value of each expression without ‘using a table or a calculator. a. cos—1(—l) b. arccos(—l/2) c. cos—1(\/2_/2) Solution 21. The value of cos—1(— 1) is the number a in [0, 7T] such that cos(a) = --1. We re— call that cos(77) = —l, and so Cos—1(-l) = 7T. b. The value of arc00s(—1/2) is the number oz in [0, 71-] such that cos(a) = -—1/2. ‘We recall that 005(277/3) = —l/2, and so arccos(-l/2) = 277/3. c. Since cos(7T/4) = Vi/Z, we have cos—1(\/§/2) = 77/4. VTRY THIS. Find the exact value of arccos(0). _ I In the next example we use a calculator to find the degree measure of an angle whose cosine isigiven. Most scientific calculators have a key labeled 003'1 that gives valuesfor the inverse cosine function. Toyget the degree measure, make sure the cal- culator is in degree mode. ...
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