Perform the operation shown below and leave the result in trigonometric form.
[8(cos10° + i sin10°)] [5(cos 200° + i sin 200°)]
[8(cos10° + i sin10°)] [5(cos 200° + i sin 200°)]
= 40(cos10° + i sin10°)(cos 200° + i sin 200°)
= 40 ( cos10cos200 + i(cos10sin200) + i(sin10cos200) + i^2 sin10sin200)
= 40( cos10cos200-sin10sin200 + i(sin10cos200 + cos10sin200) , recall i^2 = -1
= 40( cos(10+200) + i sin(10 + 200) )
= 40 (cos 210 + i sin 210)
= 40(-√3/2 - (1/2) i )
= -20√3 - 20 i
check my arithmetic
or, more directly, using deMoivre:
8cis10 * 5cis200
= 8*5 cis(10+200)
...
To perform the multiplication of complex numbers given in trigonometric form, we can use the following formula:
r₁(cosθ₁ + i sinθ₁) * r₂(cosθ₂ + i sinθ₂) = r₁r₂(cos(θ₁ + θ₂) + i sin(θ₁ + θ₂))
Let's substitute the given values:
r₁ = 8
θ₁ = 10°
r₂ = 5
θ₂ = 200°
Now we can plug in these values into the formula:
8(cos10° + i sin10°) * 5(cos200° + i sin200°) = 8 * 5 (cos(10° + 200°) + i sin(10° + 200°))
Next, we can simplify the expression inside the trigonometric functions:
cos(10° + 200°) = cos(210°)
sin(10° + 200°) = sin(210°)
We can use the trigonometric identities to find the values of cosine and sine:
cos(210°) = -√3/2
sin(210°) = -1/2
Substituting these values back into the formula:
8 * 5 (-√3/2 + i * -1/2)
Multiplying the real and imaginary parts separately:
= 40(-√3/2 + i * -1/2)
= -20√3 + 20i
Therefore, the result of the operation [8(cos10° + i sin10°)] [5(cos200° + i sin200°)] in trigonometric form is -20√3 + 20i.