How do you divide # (7-i) / (3-i) # in trigonometric form?

Question

How do you divide #  (7-i) /  (3-i)    # in trigonometric form?

Aaliyah Abdullah · Accepted Answer

In trigonometric form: #2.236(cos 0.18+i sin 0.18)# # Z=(7-i)/(3-i)# #Z=a+ib #. Modulus: #|Z|=sqrt (a^2+b^2)#; Argument:#theta=tan^-1(b/a)# Trigonometrical form : #Z =|Z|(costheta+isintheta)# #Z_1= 7- i #.Modulus:#|Z_1|=sqrt(7^2+(-1)^2) # #=sqrt 50 ~~ 7.07# Argument: #tan alpha= ((|-1|))/(|7|)# #=1/7 , alpha =tan ^-1 (1/7) ~~ 0.142, Z # lies on fourth quadrant, so #theta =2pi-alpha=2pi-0.142 ~~ 6.14# # :. Z_1=7.07(cos 6.14+i sin 6.14) #, #Z_2= 3- i #.Modulus:#|Z_2|=sqrt(3^2+(-1)^2) # #=sqrt 10 ~~ 3.16# Argument: #tan alpha= ((|-1|))/(|3|)# #=1/3 , alpha =tan ^-1 (1/3) ~~0.322, Z # lies on fourth quadrant, so #theta =2pi-alpha=2pi-0.322 ~~ 5.96# # :. Z_2=3.16(cos 5.96+i sin 5.96) #, # Z=(7-i)/(3-i)# # Z= (7.07(cos 6.14+i sin 6.14))/(3.16(cos 5.96+i sin 5.96)# #Z=2.236(cos(6.14-5.96)+isin (6.14-5.96))# or #Z=2.236(cos 0.18+i sin 0.18) =2.2+0.4 i# In trigonometric form; #2.236(cos 0.18+i sin 0.18)# [Ans]

Grace Ashcraft · Answer

undefined To divide $ \frac{7 - i}{3 - i} $ into trigonometric form:

1. Rationalize the denominator.
2. Express both the numerator and the denominator in polar form.
3. Divide the polar forms.
4. Convert the result back into rectangular form if needed.

Step 1: Rationalize the denominator:
$$ \frac{7 - i}{3 - i} \times \frac{3 + i}{3 + i} = \frac{(7 - i)(3 + i)}{(3 - i)(3 + i)} = \frac{21 + 7i - 3i - i^2}{9 + 3i - 3i - i^2} $$
$$ = \frac{21 + 4i - (-1)}{9 - (-1)} = \frac{22 + 4i}{10} $$

Step 2: Express both the numerator and the denominator in polar form:
$$ 7 - i = \sqrt{7^2 + (-1)^2} \cdot (\cos(\theta) + i \sin(\theta)) $$
$$ = \sqrt{50} \cdot (\cos(\theta) + i \sin(\theta)) $$
$$ 3 - i = \sqrt{3^2 + (-1)^2} \cdot (\cos(\phi) + i \sin(\phi)) $$
$$ = \sqrt{10} \cdot (\cos(\phi) + i \sin(\phi)) $$

Step 3: Divide the polar forms:
$$ \frac{\sqrt{50} \cdot (\cos(\theta) + i \sin(\theta))}{\sqrt{10} \cdot (\cos(\phi) + i \sin(\phi))} = \frac{\sqrt{5}}{2} \cdot \frac{(\cos(\theta) + i \sin(\theta))}{(\cos(\phi) + i \sin(\phi))} $$

Step 4: Convert the result back into rectangular form if needed:
$$ = \frac{\sqrt{5}}{2} \cdot \left(\frac{\cos(\theta)}{\cos(\phi)} + i \frac{\sin(\theta)}{\sin(\phi)}\right) $$

This is the division of $ \frac{7 - i}{3 - i} $ in trigonometric form.