Question

How do you calculate the value of $\arcsin \left( {\left. {\dfrac{1}{2}} \right)} \right.$ without a calculator?

Answer 1

Hint : First important thing about above question is that we are asked the inverse value of the sine function ie. The given $\arcsin ()$ function is actually called an inverse sine function . The inverse functions in trigonometry are those functions that give us the values of the degree at which the given trigonometric function gives the desired value given in the bracket. Thus we have to find the value of the degree at which the given function (sin in this case) gives $\dfrac{1}{2}$ on putting in a sine function. The question will be solved by the fact that we know the standard values of sine function at standard angles.

Complete step by step solution:
The given function is that of an inverse function of sine which we have to solve.
We know from standard values of sine function that sine is $\dfrac{1}{2}$ at the value of ${30^0}$ .
Thus we can write
$\arcsin \left( {\dfrac{1}{2}} \right) = {30^0}$
But this value would not be exhaustive in the set ${0^0} \leqslant \theta < {360^0}$
Another similar value will lie at
$\Rightarrow {180^0} - {30^0}$
Which is ${150^0}$
Thus another value of $\arcsin \left( {\dfrac{1}{2}} \right) = {150^0}$
Thus we get two value which satisfy the given conditions those values are ${30^0},{150^0}$
So, the correct answer is “ ${30^0},{150^0}$ ”.

Note : For given type of questions we should always remember the standard values of trigonometric functions at standard values like ${0^0},{30^0},{45^0},{60^0},{90^0}$ etc. These values will become invaluable in solving these types of questions where we have to find the solution without the use of calculators. These questions will become unsolvable without a calculator if we do not remember these values at given intervals.