Question

The value of \[\dfrac{{\cos {{30}^ \circ } + i\sin {{30}^ \circ }}}{{\cos {{60}^ \circ } - i\sin {{60}^ \circ }}}\] is
1. \[i\]
2. \[ - i\]
3. \[\dfrac{{1 + \sqrt 3 i}}{2}\]
4. \[\dfrac{{1 - \sqrt 3 i}}{2}\]

Answer 1

Hint: Firstly rationalize the denominator of the given fraction. Try to simplify it using trigonometric formulas and identities. Substitute the values for the basic trigonometric functions and operate them as required. Basic algebraic rules and trigonometric identities are to be kept in mind while solving such problems. We must know the simplification rules to solve the problem with ease.

Complete step-by-step solution:
Complex numbers have a similar definition of equality to real numbers; two complex numbers are equal if and only if both their real and imaginary parts are equal. Nonzero complex numbers written in polar form are equal if and only if they have the same magnitude and their arguments differ by an integer multiple of 2π. complex numbers are naturally thought of as existing on a two-dimensional plane.
Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every polynomial equation with real or complex coefficients has a solution which is a complex number.
We are to find the value of \[\dfrac{{\cos {{30}^ \circ } + i\sin {{30}^ \circ }}}{{\cos {{60}^ \circ } - i\sin {{60}^ \circ }}}\]
By rationalizing the given fraction we get ,
\[\dfrac{{\cos {{30}^ \circ } + i\sin {{30}^ \circ }}}{{\cos {{60}^ \circ } - i\sin {{60}^ \circ }}} \times \dfrac{{\cos {{60}^ \circ } + i\sin {{60}^ \circ }}}{{\cos {{60}^ \circ } + i\sin {{60}^ \circ }}}\]
Hence we get \[\dfrac{{\left( {\cos {{30}^ \circ } + i\sin {{30}^ \circ }} \right)\left( {\cos {{60}^ \circ } + i\sin {{60}^ \circ }} \right)}}{{\left( {{{\cos }^2}{{60}^ \circ } - {i^2}{{\sin }^2}{{60}^ \circ }} \right)}}\]
On simplifying this equation we get ,
\[\dfrac{{\cos {{30}^ \circ }\cos {{60}^ \circ } - \sin {{30}^ \circ }\sin {{60}^ \circ } + i(\sin {{30}^ \circ }\cos {{60}^ \circ } + \cos {{30}^ \circ }\sin {{60}^ \circ })}}{{\left( {{{\cos }^2}{{60}^ \circ } + {{\sin }^2}{{60}^ \circ }} \right)}}\]
On simplifying this equation we get ,
\[\left( {\dfrac{{\sqrt 3 }}{2}} \right)\left( {\dfrac{1}{2}} \right) - \left( {\dfrac{1}{2}} \right)\left( {\dfrac{{\sqrt 3 }}{2}} \right) + i\left[ {\left( {\dfrac{1}{2}} \right)\left( {\dfrac{1}{2}} \right) + \left( {\dfrac{{\sqrt 3 }}{2}} \right)\left( {\dfrac{{\sqrt 3 }}{2}} \right)} \right]\]
Which on further simplification becomes
\[0 + i\] \[ = i\]
Therefore option (1) is the correct answer.

Note: In order to solve such trigonometry questions one must have a strong grip over the concepts of trigonometry , its related formulas and rules. In addition to this we must know basic trigonometric identities so that we can simplify trigonometric expressions very easily.Rationalize the denominator of the given fraction carefully. Do the calculations carefully and recheck them so as to get the correct answer.Keep in mind that a complex number is a number that can be expressed in the form \[x + iy\] where \[x\] and \[y\] are real numbers and \[i\] (iota) is a symbol called the imaginary unit, and satisfying the equation \[{i^2} = - 1\] .