If z = 2cis 60°, find z^4 in rectangular form.
z = 2 cosine 60 + 2 i sin 60
z= 2 e^i(60)
z^4 = 16 e^i(240)
= 16 cos 240 + 16 i sin 240
= 16 (-.5 )+ 16 i(-.866 )
= -8 - 13.86 i
-8-8/3i
To find z^4, we need to raise the complex number z = 2cis(60°) to the power of 4.
Step 1: Find the magnitude (r) and argument (θ) of the complex number z:
The magnitude r is given as 2, and the argument θ is given as 60°.
Step 2: Apply De Moivre's theorem to find z^4:
De Moivre's theorem states that (r cis θ)^n = r^n cis (nθ).
In this case, n = 4, so we can apply the theorem as follows:
z^4 = (2cis(60°))^4 = 2^4 cis (4 * 60°)
Step 3: Simplify the expression:
2^4 is equal to 16, and 4 * 60° is equal to 240°.
Therefore, z^4 = 16 cis 240°.
Step 4: Convert the polar form to rectangular form:
To convert from polar form to rectangular form, we use the formulas:
x = r * cos(θ) and y = r * sin(θ).
In this case, r = 16 and θ = 240°, so we can calculate:
x = 16 * cos(240°)
y = 16 * sin(240°)
Step 5: Calculate x and y:
Using a calculator or trigonometric identities, we find that:
x = -8
y = -13.856
Step 6: Write the complex number in rectangular form:
The complex number z^4 in rectangular form is -8 - 13.856i.
To find z^4 in rectangular form, we first need to convert z from polar form to rectangular form. In polar form, z = r * cis(θ), where r is the magnitude or modulus of z, and θ is the argument or angle of z.
Given z = 2cis(60°), the magnitude r is 2, and the angle θ is 60°.
To convert from polar form to rectangular form, we use the following formulas:
x = r * cos(θ)
y = r * sin(θ)
Using these formulas, we can find the rectangular coordinates (x, y) of z:
x = 2 * cos(60°) = 2 * 0.5 = 1
y = 2 * sin(60°) = 2 * √3/2 = √3
Therefore, z = 1 + √3i in rectangular form.
To find z^4, we raise the rectangular form of z to the power 4:
z^4 = (1 + √3i)^4
To simplify this expression, we'll use the binomial theorem, which states that (a + b)^n = Σ(n choose k) * a^(n-k) * b^k, where Σ represents the sum of the terms from k = 0 to k = n.
In our case, a = 1, b = √3i, and n = 4. Applying the binomial theorem, we have:
z^4 = (1 + √3i)^4
= (4 choose 0) * 1^4 * (√3i)^0 + (4 choose 1) * 1^3 * (√3i)^1 + (4 choose 2) * 1^2 * (√3i)^2 + (4 choose 3) * 1^1 * (√3i)^3 + (4 choose 4) * 1^0 * (√3i)^4
Simplifying further:
z^4 = (1 * 1 * 1) + (4 * 1 * √3i) + (6 * 1 * -1) + (4 * 1 * -√3i) + (1 * 1 * 1)
= 1 + 4√3i - 6 - 4√3i + 1
= -4
Hence, z^4 = -4 in rectangular form.