Perform the operation shown below and leave the result in trigonometric form.
[6(cos 5pi/6 + isin 5pi/6)] [3(cos 4pi/5 + isin 4pi/5)]
6cis 5π/6 * 3cis 4π/5
= 6*3 cos(5π/6 + 4π/5)
= 18 cis 49π/30
To perform the operation shown in trigonometric form, we need to multiply the complex numbers and simplify the result.
Let's start by multiplying the real parts and imaginary parts separately.
For the real part:
6(cos 5π/6)(cos 4π/5) - 6(sin 5π/6)(sin 4π/5)
For the imaginary part:
6(sin 5π/6)(cos 4π/5) + 6(cos 5π/6)(sin 4π/5)
Now, let's simplify each part using trigonometric identities:
For the real part:
6(cos 5π/6)(cos 4π/5) - 6(sin 5π/6)(sin 4π/5)
= 6(cos(5π/6 + 4π/5))
For the imaginary part:
6(sin 5π/6)(cos 4π/5) + 6(cos 5π/6)(sin 4π/5)
= 6(sin(5π/6 + 4π/5))
Now, we have the result in trigonometric form:
6(cos(5π/6 + 4π/5) + i(sin(5π/6 + 4π/5)))
To simplify further, we can find the common denominator for the angles:
5π/6 + 4π/5 = (5π/6)(5/5) + (4π/5)(6/6)
= (25π/30) + (24π/30)
= 49π/30
Therefore, the result in trigonometric form is:
6(cos(49π/30) + i(sin(49π/30)))