There are many types of numbers in mathematics such as :
Complex Number consist of Real Number and Imaginary Number and can be expressed as :
[tex]Z = a + b ~ i[/tex]
The absolute value of complex number is also called Modulus and can be calculated using this formula :
[tex]|Z| = \sqrt { a^2 + b^2 }[/tex]
Let us tackle the problem.
De Moivre's formula for complex numbers is as follows :
[tex]\large {\boxed {[r( \cos \theta + i\sin \theta)]^n = r^n(\cos n\theta + i\sin n\theta)} }[/tex]
Using the formula above, we can solve the problem in the following way
[tex][\cos (\frac{3 \pi}{12}) + i\sin (\frac{3 \pi}{12})]^5 = \cos (5 \times \frac{3 \pi}{12}) + i\sin (5 \times \frac{3 \pi}{12})[/tex]
[tex]\large {\boxed {[\cos (\frac{3 \pi}{12}) + i\sin (\frac{3 \pi}{12})]^5 = \cos (\frac{15 \pi}{12}) + i\sin (\frac{15 \pi}{12})}}[/tex]
Grade: High School
Subject: Mathematics
Chapter: Complex Numbers
Keywords: Complex , Number , Real , Imaginary , Whole , Natural , Integers
