Evaluate Cos ⁡ 3 Π 2 \cos \frac{3 \pi}{2} Cos 2 3 Π ​ .A. 1 B. − 1 2 -\frac{1}{2} − 2 1 ​ C. 0 D. -1

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Introduction

Trigonometric functions are a crucial part of mathematics, and understanding how to evaluate them is essential for solving various mathematical problems. In this article, we will focus on evaluating the cosine function, specifically cos⁡3π2\cos \frac{3 \pi}{2}. We will explore the different methods of evaluating this function and provide a step-by-step guide on how to do it.

Understanding the Cosine Function

The cosine function is a trigonometric function that is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It is denoted by the symbol cos⁡\cos and is usually abbreviated as "cos". The cosine function is periodic, meaning that it repeats itself at regular intervals. The period of the cosine function is 2π2 \pi, which means that the function repeats itself every 2π2 \pi radians.

Evaluating cos⁡3π2\cos \frac{3 \pi}{2}

To evaluate cos⁡3π2\cos \frac{3 \pi}{2}, we need to understand the unit circle and how the cosine function is defined on it. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. The cosine function is defined as the x-coordinate of a point on the unit circle.

Step 1: Convert the Angle to Degrees

The angle 3π2\frac{3 \pi}{2} is in radians, but we can convert it to degrees by multiplying it by 180π\frac{180}{\pi}. This gives us:

3π2×180π=270∘\frac{3 \pi}{2} \times \frac{180}{\pi} = 270^\circ

Step 2: Find the Point on the Unit Circle

The point on the unit circle corresponding to the angle 270∘270^\circ is (0,−1)(0, -1). This means that the x-coordinate of the point is 0, and the y-coordinate is -1.

Step 3: Evaluate the Cosine Function

The cosine function is defined as the x-coordinate of a point on the unit circle. Therefore, we can evaluate cos⁡3π2\cos \frac{3 \pi}{2} as follows:

cos⁡3π2=cos⁡270∘=0\cos \frac{3 \pi}{2} = \cos 270^\circ = 0

Conclusion

In this article, we evaluated the cosine function at the angle 3π2\frac{3 \pi}{2}. We used the unit circle and the definition of the cosine function to find the value of cos⁡3π2\cos \frac{3 \pi}{2}. We showed that the value of cos⁡3π2\cos \frac{3 \pi}{2} is 0.

Common Mistakes to Avoid

When evaluating trigonometric functions, it is easy to make mistakes. Here are some common mistakes to avoid:

  • Not converting the angle to the correct unit: Make sure to convert the angle to the correct unit (radians or degrees) before evaluating the trigonometric function.
  • Not using the correct definition of the trigonometric function: Make sure to use the correct definition of the trigonometric function (e.g., the cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle).
  • Not considering the periodicity of the trigonometric function: Make sure to consider the periodicity of the trigonometric function (e.g., the cosine function repeats itself every 2π2 \pi radians).

Final Answer

The final answer is: 0\boxed{0}

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Q&A: Evaluating Trigonometric Functions

Q: What is the cosine function?

A: The cosine function is a trigonometric function that is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It is denoted by the symbol cos⁡\cos and is usually abbreviated as "cos".

Q: What is the period of the cosine function?

A: The period of the cosine function is 2π2 \pi, which means that the function repeats itself every 2π2 \pi radians.

Q: How do I evaluate cos⁡3π2\cos \frac{3 \pi}{2}?

A: To evaluate cos⁡3π2\cos \frac{3 \pi}{2}, you need to understand the unit circle and how the cosine function is defined on it. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. The cosine function is defined as the x-coordinate of a point on the unit circle.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It is a fundamental concept in trigonometry and is used to define the trigonometric functions.

Q: How do I find the point on the unit circle corresponding to a given angle?

A: To find the point on the unit circle corresponding to a given angle, you need to use the trigonometric functions to find the x and y coordinates of the point. For example, if the angle is θ\theta, the x-coordinate of the point is cos⁡θ\cos \theta and the y-coordinate is sin⁡θ\sin \theta.

Q: What is the value of cos⁡3π2\cos \frac{3 \pi}{2}?

A: The value of cos⁡3π2\cos \frac{3 \pi}{2} is 0.

Q: What are some common mistakes to avoid when evaluating trigonometric functions?

A: Some common mistakes to avoid when evaluating trigonometric functions include:

  • Not converting the angle to the correct unit
  • Not using the correct definition of the trigonometric function
  • Not considering the periodicity of the trigonometric function

Q: How do I use the unit circle to evaluate trigonometric functions?

A: To use the unit circle to evaluate trigonometric functions, you need to understand how the trigonometric functions are defined on the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. The trigonometric functions are defined as the x and y coordinates of a point on the unit circle.

Q: What is the significance of the unit circle in trigonometry?

A: The unit circle is a fundamental concept in trigonometry and is used to define the trigonometric functions. It is a circle with a radius of 1, centered at the origin of the coordinate plane. The trigonometric functions are defined as the x and y coordinates of a point on the unit circle.

Additional Resources

Conclusion

Evaluating trigonometric functions is an essential part of mathematics, and understanding how to do it is crucial for solving various mathematical problems. In this article, we provided a comprehensive guide on how to evaluate the cosine function, specifically cos⁡3π2\cos \frac{3 \pi}{2}. We also provided a Q&A section to help readers understand the concepts better. We hope that this article has been helpful in providing a clear and concise explanation of how to evaluate trigonometric functions.