Given Z = 2(cos 148° + isin 148°) and W = 5(cos 11° + isin 11°), find and simplify z/w.
10(cos 137° – isin 137°)
0.4(cos 137° + isin 137°)
0.4(cos 137° – isin 137°)
10(cos 137° + isin 137°)
multiply top and bottom by the conjugate of the denominator, that is, multiply top and bottom by
(cos11 - isin11)
(I will skip the ° sign)
expand and simplify to get
(2/5) (cos148cos11 - icos148sin11 + isin148cos11 - i^2sin148sin11)/(cos^2 11 - i^2 cos^2 11)
= (2/5) (cos148cos11 - cos148sin11 + i(sin148cos11 - cos148sin11)/ 1
= .4( cos(148-11) + i(sin(148-11))
= .4(cos 137 + sin137)
which is one of the choices.
To find the division of complex numbers Z and W, you need to divide the magnitude of Z by the magnitude of W, and subtract the argument of W from the argument of Z.
First, let's calculate the magnitudes of Z and W:
Magnitude of Z = |Z| = √(a^2 + b^2) = √(2^2 + 0^2) = √4 = 2
Magnitude of W = |W| = √(c^2 + d^2) = √(5^2 + 0^2) = √25 = 5
Next, let's calculate the arguments (angles) of Z and W:
Argument of Z = arg(Z) = θ = 148°
Argument of W = arg(W) = φ = 11°
Now, we can calculate the division Z/W using the formula:
Z/W = (|Z|/|W|) * [cos(θ - φ) + i*sin(θ - φ)]
Substituting the values, we get:
Z/W = (2/5) * [cos(148° - 11°) + i*sin(148° - 11°)]
= (2/5) * [cos(137°) + i*sin(137°)]
Finally, let's simplify the expression:
Z/W ≈ 0.4 * [cos(137°) + i*sin(137°)]
Therefore, the correct option is 0.4(cos 137° + isin 137°).