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

## tanθ = 2/3, so

sinθ = 2/√13
cosθ = 3/√13
z = 5(cosθ + i sinθ)
Now just finish it off

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

z = r * (cos 𝜃 + i * sin 𝜃)

where r is the magnitude of the complex number and 𝜃 is the angle in radians.

In this case, the given complex number is z = 5 cis 𝜃, where 𝜃 is the angle.

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

𝜃 = tan^(-1)(2/3)

Using a calculator, 𝜃 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 ≈ 5 * (0.857 + i * 0.516)

Multiplying both terms by 5:

z ≈ 4.285 + i * 2.58

Therefore, the complex number z in rectangular form is z ≈ 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 𝜃, we know that the magnitude (or modulus) is 5 and the angle 𝜃 is in quadrant I such that tan 𝜃 = 2/3.
2. Since tan 𝜃 = 2/3, we can use the tangent function to find the value of 𝜃. Taking the inverse tangent (arctan) of 2/3 will give us the angle 𝜃 in radians.
3. Using a calculator, find arctan(2/3) and get the value of 𝜃 in radians. Let's say 𝜃 = 𝜋/3 (approximately 1.0472 radians).
4. Now, we have the magnitude |z| = 5 and the angle 𝜃 = 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(𝜃).
- The imaginary part (y) will be given by y = |z| * sin(𝜃).
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) ≈ 5 * 0.5 ≈ 2.5
- y = 5 * sin(𝜋/3) ≈ 5 * 0.866 ≈ 4.33
Thus, the rectangular form of the complex number z = 5 cis 𝜃 is z = 2.5 + 4.33i.