An Angle Measuring $(525 N)^{\circ}$ Is In Standard Position. For Which Value Of $n$ Will The Terminal Side Fall On The \$y$[/tex\]-axis?A. $n = 2$ B. $n = 3$ C. $n = 5$ D. $n = 6$

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Introduction

In trigonometry, an angle in standard position is defined as an angle whose vertex is at the origin of a coordinate plane and whose initial side lies along the positive $x$-axis. The terminal side of the angle is the side that extends from the vertex to the point where the angle intersects the unit circle. In this article, we will explore the problem of finding the value of $n$ for which the terminal side of an angle measuring $(525 n)^{\circ}$ in standard position falls on the $y$-axis.

Understanding the Problem

To solve this problem, we need to understand the concept of angles in standard position and the unit circle. The unit circle is a circle with a radius of 1 unit, centered at the origin of the coordinate plane. The angle in standard position is measured counterclockwise from the positive $x$-axis. The terminal side of the angle intersects the unit circle at a point, which can be represented by the coordinates $(x, y)$.

The Terminal Side Falls on the $y$-Axis

For the terminal side of the angle to fall on the $y$-axis, the $x$-coordinate of the point of intersection must be 0. This means that the angle must be a multiple of $90^{\circ}$, since the $y$-axis is perpendicular to the $x$-axis and forms a right angle with it.

Finding the Value of $n$

We are given that the angle measures $(525 n)^{\circ}$. To find the value of $n$ for which the terminal side falls on the $y$-axis, we need to find the value of $n$ that makes the angle a multiple of $90^{\circ}$.

Using the Modulus Operator

We can use the modulus operator to find the remainder of the angle when divided by $360^{\circ}$. This will give us the equivalent angle in the range $(0^{\circ}, 360^{\circ})$.

Calculating the Equivalent Angle

Let's calculate the equivalent angle by taking the modulus of $(525 n)^{\circ}$ with $360^{\circ}$:

(525n)∘≡(525n)mod  360∘(525 n)^{\circ} \equiv (525 n) \mod 360^{\circ}

Simplifying the Expression

We can simplify the expression by dividing both sides by $5^{\circ}$:

(525n)∘≡(105n)mod  72∘(525 n)^{\circ} \equiv (105 n) \mod 72^{\circ}

Finding the Value of $n$

Now, we need to find the value of $n$ that makes the angle a multiple of $90^{\circ}$. We can do this by finding the value of $n$ that makes the equivalent angle a multiple of $90^{\circ}$.

Using the Modulus Operator Again

We can use the modulus operator again to find the remainder of the equivalent angle when divided by $90^{\circ}$:

(105n)mod  72∘≡(105n)mod  90∘(105 n) \mod 72^{\circ} \equiv (105 n) \mod 90^{\circ}

Simplifying the Expression Again

We can simplify the expression again by dividing both sides by $9^{\circ}$:

(105n)mod  90∘≡(11.67n)mod  10∘(105 n) \mod 90^{\circ} \equiv (11.67 n) \mod 10^{\circ}

Finding the Value of $n$

Now, we need to find the value of $n$ that makes the equivalent angle a multiple of $10^{\circ}$. We can do this by finding the value of $n$ that makes the remainder of the equivalent angle when divided by $10^{\circ}$ equal to 0.

Solving for $n$

Let's solve for $n$:

11.67n≡0mod  10∘11.67 n \equiv 0 \mod 10^{\circ}

n≡0mod  10∘n \equiv 0 \mod 10^{\circ}

n=10kn = 10k

Finding the Value of $k$

We are given that the angle measures $(525 n)^{\circ}$. We can substitute the value of $n$ into the expression:

(525n)∘=(525⋅10k)∘(525 n)^{\circ} = (525 \cdot 10k)^{\circ}

Simplifying the Expression

We can simplify the expression by multiplying both sides by $10^{\circ}$:

(525⋅10k)∘=(5250k)∘(525 \cdot 10k)^{\circ} = (5250k)^{\circ}

Finding the Value of $k$

We know that the angle measures $(525 n)^{\circ}$. We can substitute the value of $(525 n)^{\circ}$ into the expression:

(525n)∘=(5250k)∘(525 n)^{\circ} = (5250k)^{\circ}

Solving for $k$

Let's solve for $k$:

5250k=525n5250k = 525 n

k=525n5250k = \frac{525 n}{5250}

k=n10k = \frac{n}{10}

Finding the Value of $n$

We are given that the angle measures $(525 n)^{\circ}$. We can substitute the value of $k$ into the expression:

(525n)∘=(5250⋅n10)∘(525 n)^{\circ} = (5250 \cdot \frac{n}{10})^{\circ}

Simplifying the Expression

We can simplify the expression by multiplying both sides by $10^{\circ}$:

(5250⋅n10)∘=(52500⋅n10)∘(5250 \cdot \frac{n}{10})^{\circ} = (52500 \cdot \frac{n}{10})^{\circ}

Simplifying the Expression Again

We can simplify the expression again by dividing both sides by $10^{\circ}$:

(52500⋅n10)∘=(5250n)∘(52500 \cdot \frac{n}{10})^{\circ} = (5250 n)^{\circ}

Finding the Value of $n$

We know that the angle measures $(525 n)^{\circ}$. We can substitute the value of $(525 n)^{\circ}$ into the expression:

(525n)∘=(5250n)∘(525 n)^{\circ} = (5250 n)^{\circ}

Solving for $n$

Let's solve for $n$:

525n=5250n525 n = 5250 n

n=10n = 10

Conclusion

In this article, we have explored the problem of finding the value of $n$ for which the terminal side of an angle measuring $(525 n)^{\circ}$ in standard position falls on the $y$-axis. We have used the modulus operator to find the remainder of the angle when divided by $360^{\circ}$ and have simplified the expression to find the value of $n$ that makes the angle a multiple of $90^{\circ}$. We have found that the value of $n$ is 10.

Final Answer

The final answer is: 10\boxed{10}

Introduction

In our previous article, we explored the problem of finding the value of $n$ for which the terminal side of an angle measuring $(525 n)^{\circ}$ in standard position falls on the $y$-axis. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q&A

Q: What is the significance of the angle being in standard position?

A: The angle being in standard position means that the vertex of the angle is at the origin of the coordinate plane, and the initial side of the angle lies along the positive $x$-axis. This is an important concept in trigonometry, as it allows us to use the unit circle to find the values of trigonometric functions.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 unit, centered at the origin of the coordinate plane. It is used to find the values of trigonometric functions, such as sine, cosine, and tangent.

Q: How do we find the value of $n$ for which the terminal side of the angle falls on the $y$-axis?

A: To find the value of $n$, we need to find the value of $n$ that makes the angle a multiple of $90^{\circ}$. We can do this by using the modulus operator to find the remainder of the angle when divided by $360^{\circ}$.

Q: What is the modulus operator?

A: The modulus operator is a mathematical operation that finds the remainder of a division operation. In this case, we use the modulus operator to find the remainder of the angle when divided by $360^{\circ}$.

Q: How do we simplify the expression to find the value of $n$?

A: We can simplify the expression by dividing both sides by $5^{\circ}$, and then again by $9^{\circ}$, and finally by $10^{\circ}$.

Q: What is the final value of $n$?

A: The final value of $n$ is 10.

Q: Why is the value of $n$ important?

A: The value of $n$ is important because it determines the position of the terminal side of the angle. If the value of $n$ is 10, then the terminal side of the angle will fall on the $y$-axis.

Q: Can you provide an example of how to use the modulus operator to find the value of $n$?

A: Yes, here is an example:

Let's say we have an angle measuring $(525 n)^{\circ}$. We want to find the value of $n$ for which the terminal side of the angle falls on the $y$-axis. We can use the modulus operator to find the remainder of the angle when divided by $360^{\circ}$:

(525n)∘≡(525n)mod  360∘(525 n)^{\circ} \equiv (525 n) \mod 360^{\circ}

We can simplify the expression by dividing both sides by $5^{\circ}$:

(525n)mod  360∘≡(105n)mod  72∘(525 n) \mod 360^{\circ} \equiv (105 n) \mod 72^{\circ}

We can simplify the expression again by dividing both sides by $9^{\circ}$:

(105n)mod  72∘≡(11.67n)mod  10∘(105 n) \mod 72^{\circ} \equiv (11.67 n) \mod 10^{\circ}

We can simplify the expression again by dividing both sides by $10^{\circ}$:

(11.67n)mod  10∘≡(1.167n)mod  1∘(11.67 n) \mod 10^{\circ} \equiv (1.167 n) \mod 1^{\circ}

We can simplify the expression again by dividing both sides by $1^{\circ}$:

(1.167n)mod  1∘≡nmod  1∘(1.167 n) \mod 1^{\circ} \equiv n \mod 1^{\circ}

We can simplify the expression again by dividing both sides by $1^{\circ}$:

nmod  1∘≡nn \mod 1^{\circ} \equiv n

Therefore, the value of $n$ is 10.

Conclusion

In this Q&A article, we have provided additional information and clarification on the topic of finding the value of $n$ for which the terminal side of an angle measuring $(525 n)^{\circ}$ in standard position falls on the $y$-axis. We have used the modulus operator to find the remainder of the angle when divided by $360^{\circ}$ and have simplified the expression to find the value of $n$ that makes the angle a multiple of $90^{\circ}$. We have found that the value of $n$ is 10.

Final Answer

The final answer is: 10\boxed{10}