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: Recall angle on y - axis

Angles whose terminal sides fall on the \(y\) - axis are of the form \(\theta=(90 + 180k)^{\circ}\), where \(k\in\mathbb{Z}\), or equivalently, \(\theta = 90^{\circ}(2k + 1)\), \(k\in\mathbb{Z}\). Also, we know that coterminal angles are found by adding or subtracting multiples of \(360^{\circ}\). First, we can find the coterminal angle of \(525n^{\circ}\) by subtracting multiples of \(360^{\circ}\).

First, simplify \(525n^{\circ}\): \(525n^{\circ}=360n^{\circ}+165n^{\circ}\). The coterminal angle of \(525n^{\circ}\) is \(165n^{\circ}\) (after subtracting \(360n^{\circ}\)). We want this angle (or its coterminal angle) to be equal to \(90^{\circ}\) or \(270^{\circ}\) (since those are the angles on the \(y\) - axis).

Step2: Test each option

• Option \(n = 2\):

Calculate \(525\times2=1050^{\circ}\). Now, find the coterminal angle by subtracting \(2\times360^{\circ}=720^{\circ}\) from \(1050^{\circ}\). \(1050 - 720 = 330^{\circ}\), which is not on the \(y\) - axis.

• Option \(n = 3\):

Calculate \(525\times3 = 1575^{\circ}\). Subtract \(4\times360^{\circ}=1440^{\circ}\) from \(1575^{\circ}\). \(1575-1440 = 135^{\circ}\), which is not on the \(y\) - axis.

• Option \(n = 5\):

Calculate \(525\times5=2625^{\circ}\). Subtract \(7\times360^{\circ}=2520^{\circ}\) from \(2625^{\circ}\). \(2625 - 2520=105^{\circ}\), which is not on the \(y\) - axis.

• Option \(n = 6\):

Calculate \(525\times6 = 3150^{\circ}\). Subtract \(8\times360^{\circ}=2880^{\circ}\) from \(3150^{\circ}\). \(3150-2880 = 270^{\circ}\), which is an angle whose terminal side falls on the \(y\) - axis (the negative \(y\) - axis).

Another way:
We know that for the terminal side of an angle \(\theta\) to be on the \(y\) - axis, \(\theta\equiv90^{\circ}\pmod{180^{\circ}}\) (since the angles on the \(y\) - axis are \(90^{\circ},270^{\circ},450^{\circ},\cdots\) and the difference between consecutive angles on the \(y\) - axis is \(180^{\circ}\)).

So we need \(525n\equiv90\pmod{180}\)

Simplify \(525\) and \(180\) by finding their greatest common divisor. \(\gcd(525,180)=\gcd(180,525 - 2\times180)=\gcd(180,165)=\gcd(165,15)=\gcd(15,0) = 15\)

Divide the congruence \(525n\equiv90\pmod{180}\) by \(15\): \(35n\equiv6\pmod{12}\)

Since \(35\equiv - 1\pmod{12}\), the congruence becomes \(-n\equiv6\pmod{12}\), or \(n\equiv - 6\pmod{12}\), or \(n\equiv6\pmod{12}\)

Among the given options, \(n = 6\) satisfies this.

Answer:

\(n = 6\) (the option is: \(n = 6\))