Question

How do you write the trigonometric form into a complex number in standard form $3\left( \cos \left( {{18}^{\circ }}45' \right)+i\sin \left( {{18}^{\circ }}45' \right) \right)$?

Answer 1

We know that the standard form of complex numbers is equal to $a+ib$ so we are going to convert this given trigonometric form into this standard form. To convert the given trigonometric form, we are going to multiply 3 by $\cos \left( {{18}^{\circ }}45' \right)$ and then we are going to add this multiplication to the multiplication of 3 and $\sin \left( {{18}^{\circ }}45' \right)$. For this multiplication and addition to happen, we need to know the value of $\sin \left( {{18}^{\circ }}45' \right)$ and $\cos \left( {{18}^{\circ }}45' \right)$. Then substitute these values into the given complex number form.

Complete step-by-step solution:
The trigonometric form of the complex number given in the above problem is as follows:
$3\left( \cos \left( {{18}^{\circ }}45' \right)+i\sin \left( {{18}^{\circ }}45' \right) \right)$ ……….. (1)
Now, to convert the above complex number into a standard form we are going to calculate the value of $\cos \left( {{18}^{\circ }}45' \right)\And \sin \left( {{18}^{\circ }}45' \right)$ and then we substitute these values in the above expression and we get,
We know that the value of the below trigonometric ratios:
$\begin{aligned} & \cos \left( {{18}^{\circ }}45' \right)=2.8408 \\\ & \sin \left( {{18}^{\circ }}45' \right)=0.9643 \\\ \end{aligned}$
Substituting the above values in eq. (1) we get,
$\begin{aligned} & \Rightarrow 3\left( \cos \left( {{18}^{\circ }}45' \right)+i\sin \left( {{18}^{\circ }}45' \right) \right) \\\ & =3\left( 2.8408+i\left( 0.9643 \right) \right) \\\ \end{aligned}$
Multiplying 3 by first term in the bracket and also multiplying the second term written in the bracket and we get,
$\begin{aligned} & \Rightarrow 3\left( 2.8408 \right)+i\left( 3 \right)\left( 0.9643 \right) \\\ & =8.5224+i2.8929 \\\ \end{aligned}$
Hence, we have converted the given trigonometric form of complex number into standard form and that standard form is equal to $8.5224+i2.8929$.

Note: To solve the above problem, you should know how a standard complex number will look like otherwise you cannot solve this problem. Also, the conversion is not that tough because you can see that we have just written the value of $\cos \left( {{18}^{\circ }}45' \right)\And \sin \left( {{18}^{\circ }}45' \right)$ and then multiply those values with 3.