What is the rectangular form(a + bi) of:
1. 5 cis(255°)
2. sqrt3 cis (11pi/6)
5 cis(255°)
= 5(cos255° + i sin255) , note that 225 is (180 + 45)° , quadrant III
thus cos 225 = -cos45 = -√2/2
and sin 225 = -sin45 = -√2/2
5cis225
= 5(-√2/2 - √2/2 i)
= -5√2/2 - 5√2/2 i
for the second, 11π/6 is a multiple of π/6 , (or 30°)
so follow my procedure
To convert a complex number from polar form to rectangular form, we will use the formulas:
a = r * cos(θ)
b = r * sin(θ)
where r is the magnitude or modulus of the complex number, and θ is the argument or angle in radians.
1. 5 cis(255°):
First, convert the angle from degrees to radians:
θ = 255° * (π/180°) ≈ 4.4506 radians
Next, use the formulas:
a = 5 * cos(4.4506)
b = 5 * sin(4.4506)
Evaluating these expressions:
a ≈ 5 * cos(4.4506) ≈ -3.7935
b ≈ 5 * sin(4.4506) ≈ 1.8146
So, the rectangular form of 5 cis(255°) is approximately -3.7935 + 1.8146i.
2. √3 cis(11π/6):
Here, we have the angle in radians already, so we can directly use the formulas:
a = √3 * cos(11π/6)
b = √3 * sin(11π/6)
Evaluating these expressions:
a ≈ √3 * cos(11π/6) ≈ -√3/2
b ≈ √3 * sin(11π/6) ≈ -1/2
So, the rectangular form of √3 cis(11π/6) is approximately -√3/2 - 1/2i.
To find the rectangular form (a + bi) of complex numbers given in polar form (r ∠ θ), you can use the following formula:
a + bi = r(cos(θ) + i*sin(θ))
Now, let's calculate the rectangular forms for the given complex numbers:
1. For 5 cis(255°):
First, convert the angle from degrees to radians by multiplying it by π/180:
255° * (π/180) = 4.4506 radians (rounded to 4 decimal places).
Then, use the formula:
a + bi = 5*cos(4.4506) + 5*sin(4.4506)i
Calculate the cosine and sine values using a calculator or math software:
cos(4.4506) ≈ -0.4472 (rounded to 4 decimal places)
sin(4.4506) ≈ -0.8944 (rounded to 4 decimal places)
Substitute these values into the formula:
a + bi = 5*(-0.4472) + 5*(-0.8944)i
a + bi ≈ -2.236 + (-4.472)i
Therefore, the rectangular form of 5 cis(255°) is approximately -2.236 - 4.472i.
2. For √3 cis (11π/6):
First, let's simplify the angle:
11π/6 can be simplified using common factors. The largest common factor is π/6, so we divide both the numerator and denominator by π/6:
11π/6 ÷ π/6 = 11/6
Using the formula:
a + bi = √3*cos(11/6) + √3*sin(11/6)i
Calculate the cosine and sine values using a calculator or math software:
cos(11/6) ≈ 0.8660 (rounded to 4 decimal places)
sin(11/6) ≈ -0.5000 (rounded to 4 decimal places)
Substitute these values into the formula:
a + bi = √3*0.8660 + √3*(-0.5000)i
a + bi ≈ 1.499 + (-0.866)i
Therefore, the rectangular form of √3 cis (11π/6) is approximately 1.499 - 0.866i.