Figure 2 shows the graph of the sine function limited to and the graph of the cosine function limited to. We choose a domain for each function that includes the number 0. As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one. In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. The graph of each function would fail the horizontal line test. īear in mind that the sine, cosine, and tangent functions are not one-to-one functions. Recall that, for a one-to-one function, if f ( a ) = b, f ( a ) = b, then an inverse function would satisfy f − 1 ( b ) = a. In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. The following examples illustrate the inverse trigonometric functions: Be aware that sin − 1 x sin − 1 x does not mean 1 sin x. In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized in Figure 1.įor example, if f ( x ) = sin x, f ( x ) = sin x, then we would write f − 1 ( x ) = sin − 1 x. In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse.
Understanding and Using the Inverse Sine, Cosine, and Tangent Functions In this section, we will explore the inverse trigonometric functions. This is where the notion of an inverse to a trigonometric function comes into play. But what if we are given only two sides of a right triangle? We need a procedure that leads us from a ratio of sides to an angle. Find exact values of composite functions with inverse trigonometric functions.įor any right triangle, given one other angle and the length of one side, we can figure out what the other angles and sides are.Use a calculator to evaluate inverse trigonometric functions.Find the exact value of expressions involving the inverse sine, cosine, and tangent functions.Understand and use the inverse sine, cosine, and tangent functions.
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