In which quadrant of the Argand diagram does the complex number root seven times cos of negative three 𝜋 by four plus 𝑖 sin of negative three 𝜋 by four lie?
To answer this question, we’re going to need to compare our complex number with the general form of a complex number. A complex number written in polar or trigonometric form looks a little something like this. It’s 𝑧 equals 𝑟 cos 𝜃 plus 𝑖 sin 𝜃.
𝑟 is sometimes called the modulus of 𝑧. And it represents the length of the line segment that joins the point on the Argand diagram to the origin. 𝜃 is sometimes called the argument of 𝑧. This represents the angle that line segment makes with the positive real axis measured in a counterclockwise direction. Note that we sometimes give the argument in terms of the principal argument. The principal argument lies in the left-open, right-closed interval from negative 𝜋 to 𝜋. And if we have a negative angle, that indicates to us that the direction in which we’re measuring from the positive real axis is clockwise.
So let’s compare our complex number with this general form. We see that 𝑟, which is the constant factor outside of the parentheses, is the square root of seven. Then, we also see that the argument of our complex number is negative root three over four. And so, when we measure that angle from the positive real axis, we’re going to measure in a clockwise direction three 𝜋 by four radians.
Let’s plot this on our Argand diagram. We remember, of course, that the horizontal axis is the real axis and the vertical axis is the imaginary axis. A negative angle indicates to us that we measure in this direction. And we’re measuring three 𝜋 by four radians. That’s roughly here. The length of this line segment is then root seven units. And all that’s left to do is double check which quadrant this lies in. Labeling our quadrants from one through to four, and we see that the point that represents this complex number lies in the third quadrant. The answer is third.