Express 5sqrt(3) - 5i in polar form.
A: 10(cos(11pi/6) + i sin(11pi/6))
Express 4(cos(3pi/4) + i sin(3pi/4)) in rectangular form.
A: -2sqrt(2) + 2sqrt(2i
Write 2x+y=5 in polar form.
A: sqrt(5)= r cos (theta-27degrees)
Are these correct?
none of them are correct.
Let's verify the given answers one by one:
1. Express 5√3 - 5i in polar form:
To express a complex number in polar form, we need to find its magnitude (r) and argument (θ). The magnitude is given by r = √(a^2 + b^2), where a and b are the real and imaginary parts respectively.
In this case, a = 5√3 and b = -5. Thus, the magnitude is:
r = √((5√3)^2 + (-5)^2)
r = √(75 + 25)
r = √100
r = 10
To find the argument (θ), we can use the formula tan(θ) = b/a. Therefore:
tan(θ) = -5/(5√3)
tan(θ) = -1/√3
θ ≈ 11π/6 (in the fourth quadrant)
Now we can express the complex number in polar form using Euler's formula:
5√3 - 5i = 10(cos(11π/6) + i sin(11π/6))
So, the answer is "A: 10(cos(11π/6) + i sin(11π/6))."
2. Express 4(cos(3π/4) + i sin(3π/4)) in rectangular form:
To express a complex number in rectangular form, we can use the formula x = r cos(θ) and y = r sin(θ), where x and y are the real and imaginary parts respectively.
In this case, r = 4 and θ = 3π/4. Applying the formulas, we have:
x = 4 cos(3π/4) = -2√2
y = 4 sin(3π/4) = 2√2
So, the rectangular form of the complex number is -2√2 + 2√2i.
Thus, the answer is "A: -2√2 + 2√2i."
3. Write 2x + y = 5 in polar form:
The given equation is in rectangular form. Converting it to polar form involves expressing x and y in terms of the magnitude (r) and argument (θ).
We can rewrite the equation as x = r cos(θ) and y = r sin(θ). Substituting these into the equation, we have:
2(r cos(θ)) + (r sin(θ)) = 5
2rcos(θ) + rsin(θ) = 5
Now, let's factor out r and rearrange the equation:
r(2cos(θ) + sin(θ)) = 5
r = 5/(2cos(θ) + sin(θ))
To simplify this expression further, we can use trigonometric identities. In particular, we know that cos(A - B) = cos(A)cos(B) + sin(A)sin(B).
In our expression, we have 2cos(θ) + sin(θ), which can be seen as cos(A)cos(B) + sin(A)sin(B) if we set A = π/4 and B = -π/4:
2cos(θ) + sin(θ) = cos(π/4 - (-π/4))
Therefore, the expression simplifies to:
r = 5/cos(π/4 - (-π/4))
Since cos(π/4 - (-π/4)) = cos(π/2) = 0, the expression becomes:
r = 5/0
However, this expression is undefined since division by zero is not possible. Therefore, the polar form of the equation 2x + y = 5 does not exist.
The original answer you provided is incorrect.