Suppose the product of 2cis7𝜋/4 and 3cis4𝜋/5 has argument 𝑘𝜋 where 0⩽𝑘<2. Find the value of 𝑘.



We are given two complex numbers in polar form: \( 2 \text{cis} \frac{7\pi}{4} \) and \( 3 \text{cis} \frac{4\pi}{5} \), and need to find the argument \( k\pi \) of their product such that \( 0 \leq k < 2 \).

The product of two complex numbers in polar form \( r_1 \text{cis} \theta_1 \) and \( r_2 \text{cis} \theta_2 \) is given by:
\[
r_1 r_2 \text{cis} (\theta_1 + \theta_2)
\]
Applying this to our numbers:

1. The magnitudes of the numbers multiply: \( 2 \times 3 = 6 \).
2. The arguments add: \( \frac{7\pi}{4} + \frac{4\pi}{5} \).

Let's first add the arguments:
\[
\frac{7\pi}{4} + \frac{4\pi}{5} = \frac{35\pi}{20} + \frac{16\pi}{20} = \frac{51\pi}{20}
\]

Now, we need to bring the argument \( \frac{51\pi}{20} \) into the standard range \( [0, 2\pi) \). First, calculate how many full rotations of \( 2\pi \) this exceeds:
\[
\frac{51\pi}{20} = 2\pi + \frac{11\pi}{20}
\]
So, the argument in standard range is:
\[
\frac{11\pi}{20}
\]

Since the argument of the product is \( \frac{11\pi}{20} \), we can equate this to \( k\pi \). Therefore:
\[
k\pi = \frac{11\pi}{20}
\]
This gives:
\[
k = \frac{11}{20}
\]

Thus, the value of \( k \) is \( \frac{11}{20} \).

Would you like more details on any part of the process?

Here are 5 related questions to expand this problem:
1. What is the polar form of the complex number \( 2 \text{cis} \frac{3\pi}{2} \)?
2. How would you calculate the product of three complex numbers in polar form?
3. What is the geometric interpretation of multiplying two complex numbers?
4. How do you find the argument of a complex number in rectangular form?
5. How can you convert a complex number from polar form to rectangular form?

**Tip:** When adding or subtracting arguments in polar form, always reduce the result to lie within the range \( [0, 2\pi) \) or \( (-\pi, \pi] \).

Suppose the product of 2cis7𝜋/4 and 3cis4𝜋/5 has argument 𝑘𝜋 where 0⩽𝑘<2. Find the value of 𝑘.

Discover how to calculate the argument of the product of two complex numbers in polar form. This detailed solution includes the multiplication of magnitudes and the addition of arguments, followed by reducing the result to standard range [0, 2π). Suitable for Grades 10-12.

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