Convert 6 cis 135 degrees to rectangular coordiantes.
cis ... cosine , i sine
{[6 cos(135º)],[6 sin(135º)]}
To convert a complex number from polar form to rectangular form (also known as Cartesian form), we can use the following formulas:
Real Part (x-coordinate) = r * cos(θ)
Imaginary Part (y-coordinate) = r * sin(θ)
Given that the complex number is 6 cis 135 degrees, where the modulus (r) is 6 and the argument (θ) is 135 degrees, we can substitute these values into the formulas:
x-coordinate = 6 * cos(135 degrees)
y-coordinate = 6 * sin(135 degrees)
Using a calculator to evaluate the trigonometric functions, we find:
x-coordinate ≈ -4.24
y-coordinate ≈ 4.24
Therefore, the rectangular coordinates of 6 cis 135 degrees are approximately -4.24 + 4.24i.
To convert a complex number given in polar form (cis form) to rectangular form, we will need to use trigonometric functions.
The polar form of a complex number is given by r cis θ, where r is the magnitude (or absolute value) of the complex number and θ is the angle it makes with the positive real axis.
In this case, we have 6 cis 135 degrees.
Step 1: Calculate the magnitude (r)
The magnitude of the complex number is given by r = |6| = 6.
Step 2: Convert the angle to radians
To use trigonometric functions, we need to convert the angle from degrees to radians.
To convert degrees to radians, we multiply by π/180.
135 degrees * (π/180 radians/degree) = (3π/4) radians
Step 3: Use trigonometric functions to find the rectangular coordinates
The rectangular form of a complex number is given by x + yi, where x is the real part and y is the imaginary part.
x = r * cos(θ)
y = r * sin(θ)
Using the values from Step 1 and Step 2:
x = 6 * cos(3π/4) ≈ -3.54
y = 6 * sin(3π/4) ≈ 3.54
Therefore, 6 cis 135 degrees can be converted to approximately -3.54 + 3.54i in rectangular coordinates.