Select All The Correct Answers.If The Measure Of Angle \[$\theta\$\] Is \[$\frac{7 \pi}{4}\$\], Which Statements Are True?- \[$\sin (\theta)=-\frac{\sqrt{2}}{2}\$\]- The Measure Of The Reference Angle Is

Introduction

In mathematics, angles and their measures are crucial concepts in trigonometry. Understanding the properties of angles, including their measures and trigonometric functions, is essential for solving problems in various mathematical and real-world applications. In this article, we will explore the given angle measure $\frac{7 \pi}{4}$ and determine which statements are true.

Understanding the Angle Measure

The given angle measure is $\frac{7 \pi}{4}$. To understand this angle, we need to convert it to degrees. We know that $\pi$ radians is equivalent to $180^\circ$. Therefore, we can convert the angle measure as follows:

$$\frac{7 \pi}{4} = \frac{7 \cdot 180^\circ}{4} = 315^\circ$$

Analyzing the Statements

Now that we have understood the angle measure, let's analyze the given statements:

Statement 1: $\sin (\theta)=-\frac{\sqrt{2}}{2}$

To determine if this statement is true, we need to find the sine of the angle $\theta$. We know that the sine function is negative in the third and fourth quadrants. Since the angle measure $\frac{7 \pi}{4}$ is in the fourth quadrant, the sine function will be negative.

Using the unit circle or a trigonometric table, we can find that the sine of $315^\circ$ is indeed $-\frac{\sqrt{2}}{2}$. Therefore, Statement 1 is true.

Statement 2: The measure of the reference angle is $\frac{3 \pi}{4}$

The reference angle is the acute angle between the terminal side of the angle and the x-axis. To find the reference angle, we need to subtract the angle measure from $2 \pi$.

$$\text{Reference angle} = 2 \pi - \frac{7 \pi}{4} = \frac{3 \pi}{4}$$

Therefore, Statement 2 is true.

Statement 3: The measure of the reference angle is $\frac{\pi}{4}$

This statement is incorrect. As we have already found, the reference angle is $\frac{3 \pi}{4}$, not $\frac{\pi}{4}$.

Statement 4: The sine of the reference angle is $\frac{\sqrt{2}}{2}$

This statement is true. The sine of the reference angle $\frac{3 \pi}{4}$ is indeed $\frac{\sqrt{2}}{2}$.

Statement 5: The cosine of the reference angle is $-\frac{\sqrt{2}}{2}$

This statement is false. The cosine of the reference angle $\frac{3 \pi}{4}$ is actually $\frac{\sqrt{2}}{2}$, not $-\frac{\sqrt{2}}{2}$.

Conclusion

In conclusion, the true statements are:

• Statement 1: $\sin (\theta)=-\frac{\sqrt{2}}{2}$
• Statement 2: The measure of the reference angle is $\frac{3 \pi}{4}$
• Statement 4: The sine of the reference angle is $\frac{\sqrt{2}}{2}$

The false statements are:

• Statement 3: The measure of the reference angle is $\frac{\pi}{4}$
• Statement 5: The cosine of the reference angle is $-\frac{\sqrt{2}}{2}$
  Frequently Asked Questions: Understanding Angle Measures and Trigonometric Functions ====================================================================================

Introduction

In our previous article, we explored the given angle measure $\frac{7 \pi}{4}$ and determined which statements are true. In this article, we will answer some frequently asked questions related to angle measures and trigonometric functions.

Q&A

Q: What is the difference between an angle and its reference angle?

A: An angle is a measure of the amount of rotation from the initial side to the terminal side. The reference angle is the acute angle between the terminal side of the angle and the x-axis.

Q: How do I find the reference angle of an angle?

A: To find the reference angle, you need to subtract the angle measure from $2 \pi$. For example, if the angle measure is $\frac{7 \pi}{4}$, the reference angle is $2 \pi - \frac{7 \pi}{4} = \frac{3 \pi}{4}$.

Q: What is the sine of an angle?

A: The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. It can also be found using the unit circle or a trigonometric table.

Q: What is the cosine of an angle?

A: The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle. It can also be found using the unit circle or a trigonometric table.

Q: How do I determine if the sine or cosine function is positive or negative?

A: The sine function is positive in the first and second quadrants and negative in the third and fourth quadrants. The cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants.

Q: What is the difference between the sine and cosine functions?

A: The sine function is the ratio of the length of the opposite side to the length of the hypotenuse, while the cosine function is the ratio of the length of the adjacent side to the length of the hypotenuse.

Q: How do I use the unit circle to find the sine and cosine of an angle?

A: The unit circle is a circle with a radius of 1 centered at the origin. The sine and cosine of an angle can be found by drawing a line from the origin to the point on the unit circle corresponding to the angle.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• $\sin^2(x) + \cos^2(x) = 1$
• $\tan(x) = \frac{\sin(x)}{\cos(x)}$
• $\csc(x) = \frac{1}{\sin(x)}$
• $\sec(x) = \frac{1}{\cos(x)}$
• $\cot(x) = \frac{1}{\tan(x)}$

Conclusion

In conclusion, understanding angle measures and trigonometric functions is crucial for solving problems in mathematics and real-world applications. By answering these frequently asked questions, we hope to have provided a better understanding of these concepts.

Additional Resources

For further learning, we recommend the following resources:

• Khan Academy: Trigonometry
• MIT OpenCourseWare: Trigonometry
• Wolfram MathWorld: Trigonometry

We hope this article has been helpful in answering your questions about angle measures and trigonometric functions. If you have any further questions, please don't hesitate to ask.