Write z=2(\cos (5\pi )/(6)+\sin (5\pi )/(6)) in rectanguler form
To convert the given expression from polar form to rectangular form, we use the following formula:
z = r(cosθ + i sinθ)
where r is the magnitude of the complex number and θ is the angle that the number makes with the positive real axis.
In this case, we have:
r = 2 (given)
θ = 5π/6 (given)
We can now substitute these values into the formula and simplify:
z = 2(cos(5π/6) + i sin(5π/6))
z = 2(√3/2 - i/2)
z = √3 - i
Therefore, the rectangular form of z is:
z = √3 - i
AAAaannndd the bot gets it wrong yet again!
5π/6 is in QII
cos is negative in QII, and sin is positive. So the answer is
z = -√3 + i
I apologize for the mistake in my previous response. You are correct that 5π/6 is in the second quadrant (QII), where cosine is negative and sine is positive. The correct rectangular form of z is:
z = 2(cos(5π/6) + i sin(5π/6))
z = 2(-√3/2 + i/2)
z = -√3 + i
Thank you for bringing this to my attention, and please let me know if you have any further questions.
To write the complex number z = 2(cos(5π/6) + sin(5π/6)) in rectangular form, we need to convert it from polar form to rectangular form.
First, let's recall Euler's formula:
e^ix = cos(x) + i*sin(x)
Now, let's rewrite the given expression in polar form:
z = 2(cos(5π/6) + sin(5π/6))
We can convert cos(5π/6) and sin(5π/6) to their exponential forms:
cos(5π/6) = Re[e^(i * (5π/6))]
sin(5π/6) = Im[e^(i * (5π/6))]
Now, let's rewrite the expression with the exponential forms:
z = 2[ Re[e^(i * (5π/6))] + Im[e^(i * (5π/6))] ]
Using Euler's formula, we get:
z = 2[ Re[e^(i * (5π/6))] + Im[e^(i * (5π/6))] ]
= 2 * e^(i * (5π/6))
Next, let's simplify the exponential form:
e^(i * (5π/6)) = e^(i * 5π) * e^(i * π/6)
= e^(i * π) * e^(i * π/6)
= -1 * e^(i * π/6)
= -e^(i * π/6)
Therefore, z = 2 * (-e^(i * π/6))
To write it in rectangular form, we can substitute e^(i * π/6) with its rectangular form expression. The rectangular form of a complex number a + bi is simply (a, b).
Using the identity e^(ix) = cos(x) + i*sin(x), we can determine the rectangular form of -e^(i * π/6) as:
Re[-e^(i * π/6)] = -cos(π/6)
Im[-e^(i * π/6)] = -sin(π/6)
Therefore, z = 2 * (-cos(π/6) - sin(π/6))
Simplifying further, we get:
z = -2 * (cos(π/6) + sin(π/6))
Thus, the rectangular form of the complex number z is:
z = -2 * (cos(π/6) + sin(π/6))