Express the complex number 8(cos30 degrees + i sin30 degrees)sin in complex in the form a+bi
just evaluate the expression ...
8(cos30° , i sin30°)
= 8(√3/2 , i (1/2)
= 4√3 + 4i
To express the complex number in the form a+bi, we can multiply out the expression.
First, let's expand the expression 8(cos30 degrees + i sin30 degrees):
8(cos30 degrees + i sin30 degrees)
= 8cos30 degrees + 8i sin30 degrees
Using the trigonometric identity cos(theta) = sin(90 degrees - theta), we can simplify it further:
8cos30 degrees + 8i sin30 degrees
= 8cos(90 degrees - 60 degrees) + 8i sin(90 degrees - 60 degrees)
= 8cos(60 degrees) + 8i sin(60 degrees)
Since cos(60 degrees) = 1/2 and sin(60 degrees) = √3/2, we can substitute these values:
8cos(60 degrees) + 8i sin(60 degrees)
= 8(1/2) + 8i (√3/2)
= 4 + 4i√3
Now we have the complex number in the form a+bi.
Therefore, 8(cos30 degrees + i sin30 degrees) in the form a+bi is 4 + 4i√3.
To express the complex number 8(cos30 degrees + i sin30 degrees)in the form a+bi, we can start by expanding the expression.
Using the trigonometric identity for sine:
sin(θ) = (e^(iθ) - e^(-iθ)) / (2i),
where θ is the angle in radians, and e is Euler's number.
We can rewrite the complex number as:
8(cos30 degrees + i sin30 degrees) = 8e^(i30 degrees).
Now using the Euler's formula:
e^(iθ) = cosθ + i sinθ,
we replace 30 degrees with its equivalent in radians (π/6) to get:
8(cos30 degrees + i sin30 degrees) = 8e^(i(π/6)).
Using the exponential form, we can rewrite e^(i(π/6)) as:
8(cos(π/6) + i sin(π/6)).
Now, we just multiply:
8(cos(π/6) + i sin(π/6)) = 8 * cos(π/6) + 8i * sin(π/6).
Evaluating the trigonometric functions:
8 * cos(π/6) = 8 * √3/2 = 4√3,
and
8i * sin(π/6) = 8i * 1/2 = 4i.
Thus, the complex number 8(cos30 degrees + i sin30 degrees) in the form a+bi is:
4√3 + 4i.