Polar and trigonometric form of a complex number
Until now you located the affix of $z$ using two Cartesian coordinates, $(a,b)$. But a point in the plane can also be described by its distance from the origin and the angle it makes: these are the polar coordinates you already know from other contexts. Applied to $z$, that distance and that angle have their own names and greatly simplify products, quotients and powers.
Modulus and argument: the polar coordinates of $z$
Express the complex number $z = 3 + 3\sqrt{3}i$ in polar form and in trigonometric form.
Express the complex number $z = 3 + 3\sqrt{3}i$ in polar form and in trigonometric form.
Observe the representation in the complex plane of the vector defining $z$. Determine its modulus and its argument, and write its expression in polar and trigonometric…
Observe the representation in the complex plane of the vector defining $z$. Determine its modulus and its argument, and write its expression in polar and trigonometric form.
From polar to binomial: trigonometric form
If you already have $r$ and $\alpha$, recovering $a$ and $b$ is the same trigonometry as in the right-angled triangle: the adjacent side is $a = r\cos\alpha$ and the opposite side is $b = r\sin\alpha$.
Given the complex number in polar form $z = 4_{120^\circ}$, determine its expression in Cartesian form.
Given the complex number in polar form $z = 4_{120^\circ}$, determine its expression in Cartesian form.
Conjugate and opposite in polar form
In binomial form, conjugating changed the sign of $b$ and the opposite changed both signs. In polar form the idea is the same, but it shows up in the angle: conjugating reflects the argument in the real axis, and the opposite rotates it by $180^\circ$.
Find the conjugate and the opposite of the complex number $z = 6_{45^\circ}$. Express both results in polar form with positive arguments between $0^\circ$ and…
Find the conjugate and the opposite of the complex number $z = 6_{45^\circ}$. Express both results in polar form with positive arguments between $0^\circ$ and $360^\circ$.
Solving equalities in polar form
In an alternating current circuit, the voltage is represented by the complex number $V = 220_{30^\circ}$ and the impedance by $Z = 10_{-15^\circ}$. Ohm's law for…
In an alternating current circuit, the voltage is represented by the complex number $V = 220_{30^\circ}$ and the impedance by $Z = 10_{-15^\circ}$. Ohm's law for alternating current states that the current is $I = \frac{V}{Z}$. Calculate the current $I$ in polar form and subsequently express it in Cartesian form.