Sure, let's solve the problem step-by-step and express [tex]\(\left(3 \operatorname{cis} 60^\circ \right)\left(2 \operatorname{cis} 75^\circ \right)\)[/tex] in polar form.
### Step 1: Identify the polar form components
The given expressions are both in polar form, where polar form is given as [tex]\( r \operatorname{cis} \theta \)[/tex] (where "cis" stands for [tex]\(\cos \theta + i \sin \theta\)[/tex]).
- For [tex]\(3 \operatorname{cis} 60^\circ\)[/tex], the magnitude [tex]\(r_1 = 3\)[/tex] and the angle [tex]\(\theta_1 = 60^\circ\)[/tex].
- For [tex]\(2 \operatorname{cis} 75^\circ\)[/tex], the magnitude [tex]\(r_2 = 2\)[/tex] and the angle [tex]\(\theta_2 = 75^\circ\)[/tex].
### Step 2: Multiply the magnitudes
When multiplying two complex numbers in polar form, you multiply their magnitudes:
[tex]\[ R = r_1 \times r_2 = 3 \times 2 = 6 \][/tex]
### Step 3: Add the angles
Next, you add their angles:
[tex]\[ \theta = \theta_1 + \theta_2 = 60^\circ + 75^\circ = 135^\circ \][/tex]
### Step 4: Combine the results
So, when you multiply these two complex numbers in polar form, you get
[tex]\[ \left(3 \operatorname{cis} 60^\circ\right)\left(2 \operatorname{cis} 75^\circ\right) = 6 \operatorname{cis} 135^\circ \][/tex]
Thus, the product expressed in polar form is [tex]\((6, 135^\circ)\)[/tex].
