Find The Exact Value Of $\sin \frac{7 \pi}{2}$.

Introduction

In trigonometry, the sine function is a fundamental concept that plays a crucial role in solving various mathematical problems. The sine function is defined as the ratio of the length of the side opposite a given angle to the length of the hypotenuse in a right-angled triangle. However, when dealing with angles in radians, it becomes essential to understand how to evaluate the sine function for such angles. In this article, we will focus on finding the exact value of $\sin \frac{7 \pi}{2}$.

Understanding the Angle

To begin with, let's understand the angle $\frac{7 \pi}{2}$. This angle is in radians, and we can convert it to degrees to make it easier to visualize. We know that $\pi$ radians is equivalent to $180^\circ$. Therefore, $\frac{7 \pi}{2}$ radians can be converted to degrees as follows:

$$\frac{7 \pi}{2} \text{ radians} = \frac{7 \pi}{2} \times \frac{180^\circ}{\pi} = 630^\circ$$

Evaluating the Sine Function

Now that we have understood the angle, let's evaluate the sine function for $\frac{7 \pi}{2}$ radians. We know that the sine function has a periodicity of $2 \pi$, which means that the value of the sine function repeats every $2 \pi$ radians. Therefore, we can rewrite $\frac{7 \pi}{2}$ as follows:

$$\frac{7 \pi}{2} = 2 \pi + \frac{\pi}{2}$$

Using the Periodicity of the Sine Function

Since the sine function has a periodicity of $2 \pi$, we can use this property to simplify the evaluation of the sine function for $\frac{7 \pi}{2}$ radians. We know that the sine function is periodic with a period of $2 \pi$, which means that:

$$\sin (2 \pi + \theta) = \sin \theta$$

Therefore, we can rewrite the sine function for $\frac{7 \pi}{2}$ radians as follows:

$$\sin \frac{7 \pi}{2} = \sin (2 \pi + \frac{\pi}{2}) = \sin \frac{\pi}{2}$$

Evaluating the Sine Function for $\frac{\pi}{2}$

Now that we have simplified the evaluation of the sine function for $\frac{7 \pi}{2}$ radians, let's evaluate the sine function for $\frac{\pi}{2}$. We know that the sine function is defined as the ratio of the length of the side opposite a given angle to the length of the hypotenuse in a right-angled triangle. For the angle $\frac{\pi}{2}$, we have a right-angled triangle with a hypotenuse of length 1 and an opposite side of length 1. Therefore, the sine function for $\frac{\pi}{2}$ is:

$$\sin \frac{\pi}{2} = \frac{1}{1} = 1$$

Conclusion

In conclusion, we have found the exact value of $\sin \frac{7 \pi}{2}$ by using the periodicity of the sine function and evaluating the sine function for $\frac{\pi}{2}$. We have shown that $\sin \frac{7 \pi}{2} = \sin (2 \pi + \frac{\pi}{2}) = \sin \frac{\pi}{2} = 1$. This result demonstrates the importance of understanding the properties of the sine function and how to apply them to evaluate trigonometric expressions.

Additional Examples

Here are some additional examples of finding the exact value of the sine function for various angles in radians:

• $\sin \frac{3 \pi}{2} = \sin (2 \pi - \frac{\pi}{2}) = -\sin \frac{\pi}{2} = -1$
• $\sin \frac{5 \pi}{2} = \sin (2 \pi + \frac{\pi}{2}) = \sin \frac{\pi}{2} = 1$
• $\sin \frac{9 \pi}{2} = \sin (2 \pi + \frac{3 \pi}{2}) = \sin \frac{3 \pi}{2} = -1$

These examples demonstrate the importance of understanding the periodicity of the sine function and how to apply it to evaluate trigonometric expressions.

Final Thoughts

In conclusion, finding the exact value of $\sin \frac{7 \pi}{2}$ requires a deep understanding of the properties of the sine function and how to apply them to evaluate trigonometric expressions. By using the periodicity of the sine function and evaluating the sine function for $\frac{\pi}{2}$, we have shown that $\sin \frac{7 \pi}{2} = 1$. This result demonstrates the importance of understanding the properties of the sine function and how to apply them to evaluate trigonometric expressions.

Introduction

In our previous article, we discussed how to find the exact value of $\sin \frac{7 \pi}{2}$. We used the periodicity of the sine function and evaluated the sine function for $\frac{\pi}{2}$ to arrive at the result. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into finding the exact value of $\sin \frac{7 \pi}{2}$.

Q: What is the periodicity of the sine function?

A: The sine function has a periodicity of $2 \pi$, which means that the value of the sine function repeats every $2 \pi$ radians.

Q: How can we use the periodicity of the sine function to simplify the evaluation of the sine function for $\frac{7 \pi}{2}$?

A: We can rewrite $\frac{7 \pi}{2}$ as $2 \pi + \frac{\pi}{2}$ and use the periodicity of the sine function to simplify the evaluation of the sine function. This allows us to rewrite the sine function for $\frac{7 \pi}{2}$ as $\sin (2 \pi + \frac{\pi}{2}) = \sin \frac{\pi}{2}$.

Q: What is the value of the sine function for $\frac{\pi}{2}$?

A: The value of the sine function for $\frac{\pi}{2}$ is 1.

Q: How can we use the result of $\sin \frac{\pi}{2} = 1$ to find the exact value of $\sin \frac{7 \pi}{2}$?

A: We can use the result of $\sin \frac{\pi}{2} = 1$ to find the exact value of $\sin \frac{7 \pi}{2}$ by substituting $\sin \frac{\pi}{2}$ into the expression $\sin (2 \pi + \frac{\pi}{2})$. This gives us $\sin \frac{7 \pi}{2} = \sin (2 \pi + \frac{\pi}{2}) = \sin \frac{\pi}{2} = 1$.

Q: What are some additional examples of finding the exact value of the sine function for various angles in radians?

A: Here are some additional examples:

• $\sin \frac{3 \pi}{2} = \sin (2 \pi - \frac{\pi}{2}) = -\sin \frac{\pi}{2} = -1$
• $\sin \frac{5 \pi}{2} = \sin (2 \pi + \frac{\pi}{2}) = \sin \frac{\pi}{2} = 1$
• $\sin \frac{9 \pi}{2} = \sin (2 \pi + \frac{3 \pi}{2}) = \sin \frac{3 \pi}{2} = -1$

Q: How can we use the periodicity of the sine function to find the exact value of $\sin \frac{3 \pi}{2}$?

A: We can rewrite $\frac{3 \pi}{2}$ as $2 \pi - \frac{\pi}{2}$ and use the periodicity of the sine function to simplify the evaluation of the sine function. This allows us to rewrite the sine function for $\frac{3 \pi}{2}$ as $\sin (2 \pi - \frac{\pi}{2}) = -\sin \frac{\pi}{2} = -1$.

Q: What is the final answer to the problem of finding the exact value of $\sin \frac{7 \pi}{2}$?

A: The final answer to the problem of finding the exact value of $\sin \frac{7 \pi}{2}$ is 1.

Conclusion

In conclusion, finding the exact value of $\sin \frac{7 \pi}{2}$ requires a deep understanding of the properties of the sine function and how to apply them to evaluate trigonometric expressions. By using the periodicity of the sine function and evaluating the sine function for $\frac{\pi}{2}$, we have shown that $\sin \frac{7 \pi}{2} = 1$. This result demonstrates the importance of understanding the properties of the sine function and how to apply them to evaluate trigonometric expressions.

Final Thoughts

In conclusion, finding the exact value of $\sin \frac{7 \pi}{2}$ is a challenging problem that requires a deep understanding of the properties of the sine function and how to apply them to evaluate trigonometric expressions. By using the periodicity of the sine function and evaluating the sine function for $\frac{\pi}{2}$, we have shown that $\sin \frac{7 \pi}{2} = 1$. This result demonstrates the importance of understanding the properties of the sine function and how to apply them to evaluate trigonometric expressions.