Question

What is $\arcsin \left( .5 \right)$ ?

Answer 1

We need to find the value of $\arcsin \left( .5 \right)$ . We start to solve the problem by finding out the value of 0.5 in terms of angle of trigonometric function $\sin x$ . Then, we need to simplify the expression to get the desired result.

Complete step by step solution:
We need to know the basic formulae of all the trigonometric functions to solve the expression given numerically.
Sine is the trigonometric function of any specified angle that is used in the context of a right angle.
It is usually defined as the ratio of the length of the side opposite to an angle to the length of the hypotenuse of the right-angle triangle.
arcsine function is the inverse of the sine function. It is used to evaluate the angle of the sine value.
The expression of arcsinx in terms of sine function is given as follows,
$\Rightarrow \arcsin x=\dfrac{1}{\sin x}$
According to our question, we need to find the value of $\arcsin \left( .5 \right)$
We need to express the number 0.5 in terms of sinx.
$\Rightarrow \sin x=0.5$
Expressing the decimal 0.5 in terms of fraction, we get,
$\Rightarrow \sin x=\dfrac{1}{2}$
From trigonometric table,
We know that $\sin \dfrac{\pi }{6}=\dfrac{1}{2}$ . substituting the same, we get,
$\Rightarrow \sin \dfrac{\pi }{6}=\dfrac{1}{2}$
$\sin x$ on shifting to the other side of equation becomes arcsinx
Shifting the value of $\sin x$ to the other side of the equation, we get,
$\Rightarrow \dfrac{\pi }{6}=\arcsin \left( \dfrac{1}{2} \right)$
$\therefore \arcsin \left( 0.5 \right)=\dfrac{\pi }{6}$
From trigonometry, we know that the value of sinx is positive in the first and second quadrant.
The other value of arcsinx in second quadrant is given as follows,
$\Rightarrow \arcsin \left( 0.5 \right)=\pi -\dfrac{\pi }{6}$
$\Rightarrow \arcsin \left( 0.5 \right)=\dfrac{6\pi -\pi }{6}$
$\therefore \arcsin \left( 0.5 \right)=\dfrac{5\pi }{6}$

The value of $\arcsin \left( .5 \right)$ is $\dfrac{\pi }{6}$ or $\dfrac{5\pi }{6}$

Note: The inverse functions in trigonometry are also known as arc functions or anti trigonometric functions. They are majorly known as arc functions because they are most used to find the length of the arc needed to get the given or specified value. We can convert a function into an inverse function and vice versa.