731 7.5 Inverse Circular Functions The table gives all six inverse circular functions with their domains and ranges. Summary of Inverse Circular Functions Range Inverse Function Domain Interval Quadrants of the Unit Circle y = sin-1 x 3-1, 14 C -p 2 , p 2 D I and IV y = cos-1 x 3-1, 14 30, p4 I and II y = tan-1 x 1-∞, ∞2 A -p 2 , p 2 B I and IV y = cot-1 x 1-∞, ∞2 10, p2 I and II y = sec-1 x 1-∞, -14 ´ 31, ∞2 C 0, p 2 B ´ A p 2, pD I and II y = csc-1 x 1-∞, -14 ´ 31, ∞2 C -p 2, 0B ´ A0, p 2 D I and IV Inverse Function Values The inverse circular functions are formally defined with real number ranges. However, there are times when it may be convenient to find degree-measured angles equivalent to these real number values. It is also often convenient to think in terms of the unit circle and choose the inverse function values on the basis of the quadrants given in the preceding table. The inverse trigonometric function keys on a calculator give correct results for the inverse sine, inverse cosine, and inverse tangent functions. sin-10.5 = 30°, sin-11-0.52 = -30°, tan-11-12 = -45°, and cos-11-0.52 = 120° However, finding cot-1 x, sec-1 x, and csc-1 x with a calculator is not as straightforward because these functions must first be expressed in terms of tan-1 x, cos-1 x, and sin-1 x, respectively. If y = sec-1 x, for example, then we have sec y = x, which must be written in terms of cosine as follows. If sec y = x, then 1 cos y = x, or cos y = 1 x, and y = cos-1 1 x. Degree mode EXAMPLE 3 Finding Inverse Function Values (Degree-Measured Angles) Find the degree measure of u if it exists. (a) u = arctan 1 (b) u = sec-1 2 SOLUTION (a) Here u must be in 1-90°, 90°2, but because 1 is positive, u must be in quadrant I. The alternative statement, tan u = 1, leads to u = 45°. (b) Write the equation as sec u = 2. For sec-1 x, u is in quadrant I or II. Because 2 is positive, u is in quadrant I and u = 60°, since sec 60° = 2. Note that 60° Athe degree equivalent of p 3 B is in the range of the inverse secant function. S Now Try Exercises 37 and 45.
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