Question

an angle measuring (525n)° is in standard position. for which value of n will the terminal side fall on the y - axis?

n = 2
n = 3
n = 5
n = 6

Explanation:

Step1: Find the general form of angles with terminal side on \(y -\)axis

Angles with terminal side on the \(y -\)axis are of the form \((90 + 360k)^{\circ}\) or \((270+360k)^{\circ}\), \(k\in\mathbb{Z}\). In general, we can write the angle as \((90 + 180m)^{\circ}\), \(m\in\mathbb{Z}\) (since \(270=90 + 180\), and adding \(360\) is equivalent to adding \(2\times180\)). So the angle \(\theta=(525n)^{\circ}\) should satisfy \(525n=90 + 180m\) for some integers \(n\) and \(m\).

Step2: Simplify the equation \(525n=90 + 180m\)

Divide the equation \(525n=90 + 180m\) by \(15\). We get \(35n = 6+12m\), or \(35n-12m=6\). We can also use the property of angles modulo \(360\). If \(\theta=(525n)^{\circ}\), then \(\theta\bmod{360}=(525n\bmod{360})^{\circ}\). Since \(525 = 360+165\), then \(525n=(360 + 165)n\), and \(525n\bmod{360}=165n\bmod{360}\).
We want \(165n\bmod{360}=90\) or \(165n\bmod{360}=270\).
Let's check each option:

• For \(n = 2\): \(525\times2=1050\), \(1050\div360 = 2\cdots\cdots330\) (\(1050=2\times360 + 330\))
• For \(n = 3\): \(525\times3 = 1575\), \(1575\div360=4\cdots\cdots135\) (\(1575=4\times360+135\))
• For \(n = 5\): \(525\times5=2625\), \(2625\div360 = 7\cdots\cdots105\) (\(2625=7\times360 + 105\))
• For \(n = 6\): \(525\times6=3150\), \(3150\div360=8\cdots\cdots270\) (\(3150=8\times360+270\))

Answer:

\(n = 6\)