Find the product 2(cos10degree+isin 10degree).4(cos20degree+isin20degree).Then express the result in rectangular form.
From series expansions, we can deduce that
cos(x) + i sin(x)
=1 - x2/2! + x4/4! - ...
+ x - x3/3! + x5/5!
=1 + (ix)2/2! + (ix)3/3! + (ix)x4/4! + (ix)5/5! + ...
= eix
Therefore
2(cos10degree+isin 10degree).4(cos20degree+isin20degree)
=2e(iπ/18).4e(iπ/9)
=2.4e(iπ/18)+(iπ/9)
=8e(iπ/6)
=8(eiπ)(1/6)
=8(-1)(1/6)
Note:
eiπ
= cos(π) + i sin(π)
= -1 + 0
= -1
To find the product of two complex numbers and express the result in rectangular form, we can use the formula:
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
First, let's express the complex numbers in polar form:
2(cos10° + isin10°) = 2∠10°
4(cos20° + isin20°) = 4∠20°
Now, we can multiply the magnitudes and add the angles:
(2∠10°) * (4∠20°) = (2 * 4)∠(10° + 20°) = 8∠30°
To express the result in rectangular form, we convert it back to rectangular form using Euler's formula:
r∠θ = r(cosθ + isinθ)
8∠30° = 8(cos30° + isin30°)
Now, we can simplify this expression:
8(cos30° + isin30°) = 8 * cos30° + 8 * isin30° = 8 * (sqrt(3)/2 + i/2) = 4sqrt(3) + 4i
Therefore, the product of 2(cos10° + isin10°) and 4(cos20° + isin20°) expressed in rectangular form is 4sqrt(3) + 4i.