How do you simplify $ 3(\cos 4 + i\sin 4)\;.\;0.5(\cos 2.5 + i\sin 2.5) $ and express the result in rectangular form?

Hint: The given expression includes trigonometric ratios with imaginary numbers. We can use the Euler’s number to simplify the given expression. Euler’s form of complex number is given as $ {e^{ix}} = \cos x + i\sin x $ , where $ i $ is the imaginary number. Rectangular form of a complex number is written in the form of $ \left( {a + bi} \right) $ .

Complete step by step solution:
We have to simplify the given expression $ 3(\cos 4 + i\sin 4)\;.\;0.5(\cos 2.5 + i\sin 2.5) $ .
Instead of finding the cosine and sine values of the angles we will use the Euler’s form of the complex number.
Euler’s form is represented as,
  $ {e^{ix}} = \cos x + i\sin x $
where $ e $ is Euler’s number and $ i $ is the imaginary number.
We can convert both the trigonometric expressions into Euler’s form as follows,
  $
  (\cos 4 + i\sin 4) = {e^{4i}} \\
  (\cos 2.5 + i\sin 2.5) = {e^{2.5i}} \;
   $
Thus, the given expression can be written as,
  $
  3(\cos 4 + i\sin 4)\;.\;0.5(\cos 2.5 + i\sin 2.5) \\
   = 3{e^{4i}} \times \;0.5{e^{2.5i}} \;
   $
Now we can easily simplify this by using properties of exponents where the exponents are added if the bases of the multiplying numbers are common.
  $
  3{e^{4i}} \times \;0.5{e^{2.5i}} \\
   = \left( {3 \times 0.5} \right)\left( {{e^{4i}} \times {e^{2.5i}}} \right) \\
   = 1.5\left( {{e^{4i + 2.5i}}} \right) \\
   = 1.5{e^{6.5i}} \;
   $
We got our result in Euler’s form.
Now we have to convert this into rectangular form. For this we will expand the expression.
  $ 1.5{e^{6.5i}} = 1.5\left( {\cos 6.5 + i\sin 6.5} \right) = 1.5\cos 6.5 + 1.5i\sin 6.5 $
We can put this as our resulting expression after simplification.
Also, we can further move one step by finding the values of the cosine and sine of the angles (we may have to use a calculator for this purpose).
  $
  1.5\left( {\cos 6.5 + i\sin 6.5} \right) \\
   \approx 1.5\left( {0.977 + 0.215i} \right) \\
   = 1.466 + 0.323i \;
   $
Hence, the simplified form of the given expression is $ 1.5\left( {\cos 6.5 + i\sin 6.5} \right) \approx 1.466 + 0.323i $ .
So, the correct answer is “ $ 1.5\left( {\cos 6.5 + i\sin 6.5} \right) \approx 1.466 + 0.323i $ ”.

Note: We used Euler’s form of complex number to simplify the given expression. We could have also used trigonometric identities to arrive at the same result. But when an approach is not specified in the question it is better to use a simple approach. Calculation of the final value of cosine and sine is optional as it may require a calculator.