Question

How do you find the exact value of \[\arcsin \left( { - \dfrac{1}{2}} \right)\].

Answer 1

Hint: he $\arcsin x$ is the inverse trigonometric function of $\sin x$ . $\operatorname{Sin} $ Function is a periodic function with a period of $2\pi $ , the domain of $\sin $ function is defined in the interval of $\left( { - \infty ,\infty } \right)$ and the range of$\sin $function is $\left[ { - 1,1} \right]$ . $\operatorname{Sin} $ is an odd function which means its graph will be symmetric about origin. An odd function is defined as a function which is symmetric about the origin. The mathematical expression for odd function is defined as $f\left( { - x} \right) + f\left( x \right) = 0$ , the $x - $intercept of $\sin $function is $n\pi $ ,where $k$ is an integer and the $y - $ intercept of the function is 0 .The maximum points of the function are $\left( {\dfrac{\pi }{2} + 2n\pi ,1} \right)$ where $n$ is an integer and the minimum points of the function are $\left( {\dfrac{{3\pi }}{2} + 2n\pi , - 1} \right)$ where $k$ is an integer. To find the exact value of inverse trigonometric function we will use trigonometric ratios for standard angles. The range of the $\arcsin $ or ${\sin ^{ - 1}}x$ is $\left( {\dfrac{\pi }{2}, - \dfrac{\pi }{2}} \right)$ . The $\arcsin $ of a positive value is found in first quadrant or zero to $\dfrac{\pi }{2}$ and $\arcsin $ of a negative value is zero to $ - \dfrac{\pi }{2}$ .

Complete step by step answer:
Step: 1 consider the given function,
$\arcsin $$\left( { - \dfrac{1}{2}} \right) = $$x$
Use the inverse trigonometry properties to write the equation in inverse form.
$ \Rightarrow {\sin ^{ - 1}}\left( { - \dfrac{1}{2}} \right) = x$
Take sine function to the both side of equation,
$\sin \left( {{{\sin }^{ - 1}}\left( { - \dfrac{1}{2}} \right)} \right) = \sin x$
Step: 2 use the property of inverse trigonometry to solve the equation,
$ \Rightarrow \sin x = - \dfrac{1}{2}$
Since the sine function is odd function,
$ \Rightarrow \sin \left( { - x} \right) = - \sin x$
$
   \Rightarrow \sin x = \sin \left( { - {{30}^ \circ }} \right) \\
   \Rightarrow x = - {30^ \circ } \\
 $
Step: 3 convert the degree form of angle into radian by multiplying $\dfrac{\pi }{{180}}$.
$
   \Rightarrow x = - 30 \times \dfrac{\pi }{{180}} \\
   \Rightarrow x = - \dfrac{\pi }{6} \\
 $
Therefore, the exact value of the inverse trigonometric function is $ - \dfrac{\pi }{6}$

Note: The inverse of a trigonometric function is represented as the arc of the trigonometric function. To find the exact value of inverse trigonometric function, use the standard value of the trigonometric ratios. Convert the degree of angle into radian by multiplying with $\dfrac{\pi }{{180}}$ .