Question

If $z = i\log (2 - \sqrt 3 )$ , then $\cos z$ is equal to
$1)i$
$2)2i$
$3)1$
$4)2$

Answer 1

Hint: First, complex numbers are the real and imaginary combined numbers as in the form of $z = x + iy$, where x and y are the real numbers and $i$ is the imaginary.
Imaginary $i$ can be also represented into the real values only if, ${i^2} = - 1$
Since we have given the value of the z, then we need to find the value of $\cos z$ using the given information.

Formula used: $\cos z = \dfrac{{{e^{iz}} + {e^{ - iz}}}}{2}$

Complete step-by-step solution:
Since from the given that we have $z = i\log (2 - \sqrt 3 )$ also the trigonometry cosine can be expressed as the exponent of $\cos z = \dfrac{{{e^{iz}} + {e^{ - iz}}}}{2}$
Now just substitute the value of the z, in the cosine then we have $\cos z = \dfrac{{{e^{iz}} + {e^{ - iz}}}}{2} \Rightarrow \cos z = \dfrac{{{e^{i(i\log (2 - \sqrt 3 ))}} + {e^{ - i(i\log (2 - \sqrt 3 ))}}}}{2}$
Since we know that ${i^2} = - 1$ then we get $\cos z = \dfrac{{{e^{i(i\log (2 - \sqrt 3 ))}} + {e^{ - i(i\log (2 - \sqrt 3 ))}}}}{2} \Rightarrow \cos z = \dfrac{{{e^{ - \log (2 - \sqrt 3 )}} + {e^{\log (2 - \sqrt 3 )}}}}{2}$
Since exponent and the logarithm is the inverse process and they can be represented as ${e^{\log (a)}} = a,{e^{ - \log (a)}} = \dfrac{1}{a}$
Thus, by this concept, we have the above equation as $\cos z = \dfrac{{{e^{ - \log (2 - \sqrt 3 )}} + {e^{\log (2 - \sqrt 3 )}}}}{2} \Rightarrow \cos z = \dfrac{{\dfrac{1}{{(2 - \sqrt 3 )}} + (2 - \sqrt 3 )}}{2}$
Now cross multiplying we get $\cos z = \dfrac{{\dfrac{{1 + {{(2 - \sqrt 3 )}^2}}}{{(2 - \sqrt 3 )}}}}{2}$ and further soling we have $\cos z = \dfrac{{\dfrac{{1 + {{(2 - \sqrt 3 )}^2}}}{{(2 - \sqrt 3 )}}}}{2} \Rightarrow \cos z = \dfrac{{\dfrac{{4 + 4 - 4\sqrt 3 }}{{(2 - \sqrt 3 )}}}}{2} \Rightarrow \dfrac{{4((2 - \sqrt 3 ))}}{{(2 - \sqrt 3 )}} \times \dfrac{1}{2}$
Hence canceling the common terms, we have $\cos z = \dfrac{{4((2 - \sqrt 3 ))}}{{(2 - \sqrt 3 )}} \times \dfrac{1}{2} \Rightarrow \cos z = 4 \times \dfrac{1}{2} \Rightarrow \cos z = 2$
Therefore, the option $4)2$ is correct.

Note: We will first understand what the logarithmic operator represents in mathematics. A logarithm function or log operator is used when we have to deal with the powers of a number, to understand it better which is $\log {x^m} = m\log x$
The conjugate of a complex number represents the reflection of that complex number about the real axis on the argand plane.
When the imaginary $i$ of the complex number is replaced with $ - i$ , we get the conjugate of that complex number that shows the image of the particular complex number about the plane.
The logarithm function we used $\log {x^m} = m\log x$ and logarithm derivative function can be represented as $\dfrac{d}{{dx}}\log x = \dfrac{1}{x}$
And the exponent of the logarithm is ${e^{\log (a)}} = a,{e^{ - \log (a)}} = \dfrac{1}{a}$