Which Of The Following Best Explains Why Cos ⁡ 2 Π 3 ≠ Cos ⁡ 5 Π 3 \cos \frac{2 \pi}{3} \neq \cos \frac{5 \pi}{3} Cos 3 2 Π ​  = Cos 3 5 Π ​ ?A. The Angles Do Not Have The Same Reference Angle.B. Cosine Is Negative In The Second Quadrant And Positive In The Fourth Quadrant.C. Cosine Is Positive

Understanding the Cosine Function: A Key to Solving the Mystery of $\cos \frac{2 \pi}{3} \neq \cos \frac{5 \pi}{3}$

The cosine function is a fundamental concept in mathematics, particularly in trigonometry. It is used to describe the relationship between the angles and side lengths of triangles. However, the cosine function can be complex, and its behavior can be counterintuitive. In this article, we will explore the concept of the cosine function and use it to explain why $\cos \frac{2 \pi}{3} \neq \cos \frac{5 \pi}{3}$.

The cosine function is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The cosine function can be positive or negative, depending on the quadrant in which the angle lies.

A reference angle is an angle between 0 and 90 degrees that has the same absolute value as the given angle. Reference angles are used to determine the sign of the cosine function. If the reference angle is between 0 and 90 degrees, the cosine function is positive. If the reference angle is between 90 and 180 degrees, the cosine function is negative.

The Angles $\frac{2 \pi}{3}$ and $\frac{5 \pi}{3}$

The angles $\frac{2 \pi}{3}$ and $\frac{5 \pi}{3}$ are both coterminal angles, meaning that they have the same terminal side. However, they have different reference angles.

• The reference angle for $\frac{2 \pi}{3}$ is $\frac{\pi}{3}$, which is between 0 and 90 degrees. Therefore, the cosine function is positive for this angle.
• The reference angle for $\frac{5 \pi}{3}$ is also $\frac{\pi}{3}$, but it lies in the fourth quadrant. Therefore, the cosine function is negative for this angle.

Why $\cos \frac{2 \pi}{3} \neq \cos \frac{5 \pi}{3}$

The reason why $\cos \frac{2 \pi}{3} \neq \cos \frac{5 \pi}{3}$ is that the cosine function has different signs for these two angles. The cosine function is positive for $\frac{2 \pi}{3}$, but negative for $\frac{5 \pi}{3}$. This is because the reference angles for these two angles are the same, but they lie in different quadrants.

In conclusion, the cosine function is a complex function that can be positive or negative, depending on the quadrant in which the angle lies. The reference angle is a key concept in understanding the behavior of the cosine function. By using reference angles, we can determine the sign of the cosine function and explain why $\cos \frac{2 \pi}{3} \neq \cos \frac{5 \pi}{3}$.

• The cosine function is a periodic function that oscillates between -1 and 1.
• The cosine function can be positive or negative, depending on the quadrant in which the angle lies.
• Reference angles are used to determine the sign of the cosine function.
• The angles $\frac{2 \pi}{3}$ and $\frac{5 \pi}{3}$ have the same reference angle, but lie in different quadrants.
• The cosine function is positive for $\frac{2 \pi}{3}$, but negative for $\frac{5 \pi}{3}$.

• What is the reference angle for $\frac{2 \pi}{3}$?
  • The reference angle for $\frac{2 \pi}{3}$ is $\frac{\pi}{3}$.
• What is the reference angle for $\frac{5 \pi}{3}$?
  • The reference angle for $\frac{5 \pi}{3}$ is also $\frac{\pi}{3}$.
• Why is the cosine function positive for $\frac{2 \pi}{3}$?
  • The cosine function is positive for $\frac{2 \pi}{3}$ because the reference angle lies between 0 and 90 degrees.
• Why is the cosine function negative for $\frac{5 \pi}{3}$?
  • The cosine function is negative for $\frac{5 \pi}{3}$ because the reference angle lies between 90 and 180 degrees.
    Cosine Function Q&A: Understanding the Behavior of the Cosine Function

The cosine function is a fundamental concept in mathematics, particularly in trigonometry. It is used to describe the relationship between the angles and side lengths of triangles. However, the cosine function can be complex, and its behavior can be counterintuitive. In this article, we will explore the concept of the cosine function and answer some frequently asked questions about it.

Q: What is the cosine function?

A: The cosine function is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

Q: What is the reference angle for an angle?

A: The reference angle is an angle between 0 and 90 degrees that has the same absolute value as the given angle. Reference angles are used to determine the sign of the cosine function.

Q: Why is the cosine function positive for some angles and negative for others?

A: The cosine function is positive for angles that lie between 0 and 90 degrees, and negative for angles that lie between 90 and 180 degrees.

Q: What is the difference between the angles $\frac{2 \pi}{3}$ and $\frac{5 \pi}{3}$?

A: The angles $\frac{2 \pi}{3}$ and $\frac{5 \pi}{3}$ are both coterminal angles, meaning that they have the same terminal side. However, they have different reference angles.

Q: Why is the cosine function positive for $\frac{2 \pi}{3}$ and negative for $\frac{5 \pi}{3}$?

A: The cosine function is positive for $\frac{2 \pi}{3}$ because the reference angle lies between 0 and 90 degrees. The cosine function is negative for $\frac{5 \pi}{3}$ because the reference angle lies between 90 and 180 degrees.

Q: Can you give an example of how to use the cosine function to solve a problem?

A: Yes, here is an example:

Suppose we want to find the cosine of the angle $\frac{3 \pi}{4}$. We can use the reference angle to determine the sign of the cosine function. The reference angle for $\frac{3 \pi}{4}$ is $\frac{\pi}{4}$, which lies between 0 and 90 degrees. Therefore, the cosine function is positive for this angle. The cosine of $\frac{3 \pi}{4}$ is equal to the cosine of $\frac{\pi}{4}$, which is $\frac{\sqrt{2}}{2}$.

Q: What are some common mistakes to avoid when working with the cosine function?

A: Some common mistakes to avoid when working with the cosine function include:

• Not using the reference angle to determine the sign of the cosine function
• Not considering the quadrant in which the angle lies
• Not using the correct formula for the cosine function

In conclusion, the cosine function is a complex function that can be positive or negative, depending on the quadrant in which the angle lies. By using reference angles, we can determine the sign of the cosine function and solve problems involving the cosine function. We hope that this article has helped to clarify some of the common questions and misconceptions about the cosine function.

• The cosine function is a periodic function that oscillates between -1 and 1.
• The cosine function can be positive or negative, depending on the quadrant in which the angle lies.
• Reference angles are used to determine the sign of the cosine function.
• The angles $\frac{2 \pi}{3}$ and $\frac{5 \pi}{3}$ have the same reference angle, but lie in different quadrants.
• The cosine function is positive for $\frac{2 \pi}{3}$, but negative for $\frac{5 \pi}{3}$.

• What is the reference angle for $\frac{3 \pi}{4}$?
  • The reference angle for $\frac{3 \pi}{4}$ is $\frac{\pi}{4}$.
• What is the cosine of $\frac{3 \pi}{4}$?
  • The cosine of $\frac{3 \pi}{4}$ is equal to the cosine of $\frac{\pi}{4}$, which is $\frac{\sqrt{2}}{2}$.
• Why is the cosine function positive for $\frac{3 \pi}{4}$?
  • The cosine function is positive for $\frac{3 \pi}{4}$ because the reference angle lies between 0 and 90 degrees.
• What are some common mistakes to avoid when working with the cosine function?
  • Some common mistakes to avoid when working with the cosine function include not using the reference angle to determine the sign of the cosine function, not considering the quadrant in which the angle lies, and not using the correct formula for the cosine function.