write the complex number in standard form a+bi
sqrt 7 (cos pi/12 + i sin pi/12)
(sqrt = square root, i= imaginary number)
To write the given complex number in standard form a + bi, we can use the trigonometric form of a complex number. Given the expression:
sqrt(7) * (cos(pi/12) + i sin(pi/12))
We can rewrite it as:
sqrt(7) * cos(pi/12) + i * sqrt(7) * sin(pi/12)
Now, we need to simplify the expression using the trigonometric identities. The cosine of pi/12 can be expressed as sqrt(6 + sqrt(3))/4, and the sine of pi/12 can be expressed as sqrt(6 - sqrt(3))/4.
Substituting these values, the expression becomes:
sqrt(7) * (sqrt(6 + sqrt(3))/4) + i * sqrt(7) * (sqrt(6 - sqrt(3))/4)
We can further simplify by multiplying the numerator and denominator of each term:
(sqrt(7) * sqrt(6 + sqrt(3)) + i * sqrt(7) * sqrt(6 - sqrt(3))) / 4
To simplify the square root terms, we multiply them together:
sqrt(7 * (6 + sqrt(3))) + i * sqrt(7 * (6 - sqrt(3)))
Now we can distribute and rearrange:
(sqrt(42 + 7sqrt(3)) + i * sqrt(42 - 7sqrt(3))) / 4
Thus, the complex number sqrt(7) * (cos(pi/12) + i sin(pi/12)) in standard form a + bi is:
(sqrt(42 + 7sqrt(3))) / 4 + (sqrt(42 - 7sqrt(3))) / 4 * i
Note that this complex number cannot be simplified further in terms of common radicals.