In the previous chapter, we worked with trigonometry on a right triangle to solve for the sides of a triangle given one side and an additional angle. These may be labeled, for example, SIN −1 −1, ARCSIN, or ASIN. Most scientific calculators and calculator-emulating applications have specific keys or buttons for the inverse sine, cosine, and tangent functions. To evaluate inverse trigonometric functions that do not involve the special angles discussed previously, we will need to use a calculator or other type of technology. Using a Calculator to Evaluate Inverse Trigonometric Functions The correct angle is tan − 1 ( 1 ) = π 4.

ⓓ Evaluating tan − 1 ( 1 ), tan − 1 ( 1 ), we are looking for an angle in the interval ( − π 2, π 2 ) ( − π 2, π 2 ) with a tangent value of 1.The angle that satisfies this is cos − 1 ( − 3 2 ) = 5 π 6. ⓒTo evaluate cos − 1 ( − 3 2 ), cos − 1 ( − 3 2 ), we are looking for an angle in the interval with a cosine value of − 3 2.For that, we need the negative angle coterminal with 7 π 4 : 7 π 4 : sin − 1 ( − 2 2 ) = − π 4. ⓑ To evaluate sin − 1 ( − 2 2 ), sin − 1 ( − 2 2 ), we know that 5 π 4 5 π 4 and 7 π 4 7 π 4 both have a sine value of − 2 2, − 2 2, but neither is in the interval.Remember that the inverse is a function, so for each input, we will get exactly one output.

In other words, what angle x x would satisfy sin ( x ) = 1 2 ? sin ( x ) = 1 2 ? There are multiple values that would satisfy this relationship, such as π 6 π 6 and 5 π 6, 5 π 6, but we know we need the angle in the interval, , so the answer will be sin − 1 ( 1 2 ) = π 6.

ⓐ Evaluating sin − 1 ( 1 2 ) sin − 1 ( 1 2 ) is the same as determining the angle that would have a sine value of 1 2.

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.

Trigonometry compositions calculator