Question

find the square root of -25 that graphs on the positive y-axis. ?(cos □° + i sin □°)

Explanation:

Step1: Recall Complex Number Polar Form

A complex number \( z = r(\cos\theta + i\sin\theta) \), where \( r \) is the modulus and \( \theta \) is the argument. For \( -25 \), first write it as a complex number: \( -25 + 0i \). The modulus \( r=\sqrt{(-25)^2 + 0^2}=25 \).

Step2: Find the Argument for Positive y - axis

We want the square root that graphs on the positive y - axis. The original number \( -25 \) is on the negative x - axis, with argument \( 180^\circ \). When we take the square root of a complex number in polar form \( z = r(\cos\theta + i\sin\theta) \), the square roots have modulus \( \sqrt{r} \) and arguments \( \frac{\theta + 360^\circ k}{2} \), \( k = 0,1 \). For \( z=-25 = 25(\cos180^\circ+i\sin180^\circ) \), the square roots:

• For \( k = 0 \): \( \theta_1=\frac{180^\circ}{2}=90^\circ \)
• For \( k = 1 \): \( \theta_2=\frac{180^\circ + 360^\circ}{2}=270^\circ \)

The square root with argument \( 90^\circ \) (positive y - axis) has modulus \( \sqrt{25}=5 \). So the square root is \( 5(\cos90^\circ + i\sin90^\circ) \).

Answer:

The first box is \( 5 \), the second box is \( 90 \), the third box is \( 90 \). So the answer is \( 5(\cos90^{\circ}+i\sin90^{\circ}) \) with the first blank being \( 5 \), the second blank being \( 90 \), and the third blank being \( 90 \).