# There is an angle π in quadrant I such that tan π = 2/3. Convert the complex number z = 5 cis π from polar to rectangular form.

## tanΞΈ = 2/3, so

sinΞΈ = 2/β13
cosΞΈ = 3/β13
z = 5(cosΞΈ + i sinΞΈ)
Now just finish it off

## To convert a complex number from polar form to rectangular form, we use the following formulas:

z = r * (cos π + i * sin π)

where r is the magnitude of the complex number and π is the angle in radians.

In this case, the given complex number is z = 5 cis π, where π is the angle.

We already know that tan π = 2/3. To find π, we can use the inverse tangent function:

π = tan^(-1)(2/3)

Using a calculator, π is approximately 0.588003 radians.

Now we can convert the complex number to rectangular form:

z = 5 * (cos(0.588003) + i * sin(0.588003))

We can use the values from the trigonometric functions to evaluate this expression:

z β 5 * (0.857 + i * 0.516)

Multiplying both terms by 5:

z β 4.285 + i * 2.58

Therefore, the complex number z in rectangular form is z β 4.285 + 2.58i.

## To convert a complex number from polar form to rectangular form, we can use the following formulas:

The rectangular form of a complex number is z = x + yi, where x is the real part and y is the imaginary part.

1. Given z = 5 cis π, we know that the magnitude (or modulus) is 5 and the angle π is in quadrant I such that tan π = 2/3.
2. Since tan π = 2/3, we can use the tangent function to find the value of π. Taking the inverse tangent (arctan) of 2/3 will give us the angle π in radians.
3. Using a calculator, find arctan(2/3) and get the value of π in radians. Let's say π = π/3 (approximately 1.0472 radians).
4. Now, we have the magnitude |z| = 5 and the angle π = π/3. We can use these values to calculate the real and imaginary parts of the complex number z.
- The real part (x) will be given by x = |z| * cos(π).
- The imaginary part (y) will be given by y = |z| * sin(π).
5. Plug in the values: x = 5 * cos(π/3) and y = 5 * sin(π/3) into the equations.
6. Calculate:
- x = 5 * cos(π/3) β 5 * 0.5 β 2.5
- y = 5 * sin(π/3) β 5 * 0.866 β 4.33
Thus, the rectangular form of the complex number z = 5 cis π is z = 2.5 + 4.33i.