write the complex number in standard form a+bi
sqrt 7 (cos pi/12 + i sin pi/12)
(sqrt = square root, i= imaginary number)
check condition of problem
what do you mean?
Maybe we must find the square of
sqrt7(cosPi/12+isinPi/12)?
sqrt7=2.646
cosPi/12=0.966
sinPi/12=0.259
2.646(0.966+i0.259)=2.556+0.685i
thank you
To write the complex number in standard form a+bi, we can use the trigonometric form of complex numbers. The given complex number is sqrt 7 (cos pi/12 + i sin pi/12).
In the trigonometric form of a complex number, we can express it as r(cosθ + isinθ), where r represents the magnitude (distance from the origin) and θ represents the angle with the positive x-axis.
We are given that the magnitude is sqrt 7, which means r = sqrt 7.
The angle θ is pi/12.
To convert this trigonometric form into standard form a+bi, we need to use the trigonometric identities:
cosθ = cos(pi/12) = (√3 + 1) / 2
sinθ = sin(pi/12) = (√3 - 1) / 2
Let's substitute the values of cos(pi/12) and sin(pi/12) into the equation:
sqrt 7 (cos pi/12 + i sin pi/12) = sqrt 7 * ((√3 + 1) / 2 + i * (√3 - 1) / 2)
= sqrt 7 * (√3 + 1) / 2 + sqrt 7 * i * (√3 - 1) / 2
= sqrt(7 * (√3 + 1)) / 2 + sqrt(7 * (√3 - 1)) / 2 * i
Simplifying further, we get:
= (sqrt(7√3) + sqrt(7)) / 2 + [(sqrt(7√3) - sqrt(7)) / 2] * i
Hence, the complex number in standard form a + bi is:
(a) = (sqrt(7√3) + sqrt(7)) / 2
(b) = (sqrt(7√3) - sqrt(7)) / 2
Therefore, the complex number in standard form is:
(sqrt(7√3) + sqrt(7)) / 2 + [(sqrt(7√3) - sqrt(7)) / 2] * i