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Evaluate each of the following.
a. $\text{\hspace{0.17em}}-\frac{\pi}{2};\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}-\frac{\pi}{4};\text{\hspace{0.17em}}$ c. $\text{\hspace{0.17em}}\pi ;\text{\hspace{0.17em}}$ d. $\text{\hspace{0.17em}}\frac{\pi}{3}\text{\hspace{0.17em}}$
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. Most scientific calculators and calculator-emulating applications have specific keys or buttons for the inverse sine, cosine, and tangent functions. These may be labeled, for example, SIN-1, ARCSIN, or ASIN.
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. Using the inverse trigonometric functions, we can solve for the angles of a right triangle given two sides, and we can use a calculator to find the values to several decimal places.
In these examples and exercises, the answers will be interpreted as angles and we will use $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ as the independent variable. The value displayed on the calculator may be in degrees or radians, so be sure to set the mode appropriate to the application.
Evaluate $\text{\hspace{0.17em}}{\mathrm{sin}}^{-1}(0.97)\text{\hspace{0.17em}}$ using a calculator.
Because the output of the inverse function is an angle, the calculator will give us a degree value if in degree mode and a radian value if in radian mode. Calculators also use the same domain restrictions on the angles as we are using.
In radian mode, $\text{\hspace{0.17em}}{\mathrm{sin}}^{-1}(0.97)\approx \mathrm{1.3252.}\text{\hspace{0.17em}}$ In degree mode, $\text{\hspace{0.17em}}{\mathrm{sin}}^{-1}(0.97)\approx \mathrm{75.93\xb0.}\text{\hspace{0.17em}}$ Note that in calculus and beyond we will use radians in almost all cases.
Evaluate $\text{\hspace{0.17em}}{\mathrm{cos}}^{-1}\left(-0.4\right)\text{\hspace{0.17em}}$ using a calculator.
1.9823 or 113.578°
Given two sides of a right triangle like the one shown in [link] , find an angle.
Solve the triangle in [link] for the angle $\text{\hspace{0.17em}}\theta .$
Because we know the hypotenuse and the side adjacent to the angle, it makes sense for us to use the cosine function.
Solve the triangle in [link] for the angle $\text{\hspace{0.17em}}\theta .$
${\mathrm{sin}}^{\mathrm{-1}}(0.6)=\mathrm{36.87\xb0}=0.6435\text{\hspace{0.17em}}$ radians
There are times when we need to compose a trigonometric function with an inverse trigonometric function. In these cases, we can usually find exact values for the resulting expressions without resorting to a calculator. Even when the input to the composite function is a variable or an expression, we can often find an expression for the output. To help sort out different cases, let $\text{\hspace{0.17em}}f(x)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g(x)\text{\hspace{0.17em}}$ be two different trigonometric functions belonging to the set $\text{\hspace{0.17em}}\left\{\mathrm{sin}(x),\mathrm{cos}(x),\mathrm{tan}(x)\right\}\text{\hspace{0.17em}}$ and let $\text{\hspace{0.17em}}{f}^{-1}(y)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{g}^{-1}(y)$ be their inverses.
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