What Is The Maximum Of $f(x) = \sin(x)$?A. − 2 Π -2\pi − 2 Π B. − 1 -1 − 1 C. 1 1 1 D. 2 Π 2\pi 2 Π

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Introduction

The sine function, denoted by $\sin(x)$, is a fundamental concept in trigonometry and mathematics. It is a periodic function that oscillates between $-1$ and $1$. In this article, we will explore the maximum value of the sine function, $f(x) = \sin(x)$.

Understanding the Sine Function

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. It is a periodic function with a period of $2\pi$, meaning that it repeats itself every $2\pi$ radians. The sine function can be represented graphically as a wave that oscillates between $-1$ and $1$.

Maximum Value of the Sine Function

To find the maximum value of the sine function, we need to consider its graph. The graph of the sine function is a wave that oscillates between $-1$ and $1$. The maximum value of the sine function occurs when the graph reaches its peak, which is at $1$. This is because the sine function is an odd function, meaning that it is symmetric about the origin. As a result, the maximum value of the sine function is $1$.

Mathematical Proof

To prove that the maximum value of the sine function is $1$, we can use the following mathematical argument:

Let $x$ be any real number. Then, we can write:

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

Using the angle addition formula for sine, we get:

$$\sin(x) = \cos(x - \frac{\pi}{2})$$

Since the cosine function is an even function, we can write:

$$\cos(x - \frac{\pi}{2}) = \cos(-\frac{\pi}{2} + x)$$

Using the fact that the cosine function is an even function, we get:

$$\cos(-\frac{\pi}{2} + x) = \cos(\frac{\pi}{2} - x)$$

Since the cosine function is an even function, we can write:

$$\cos(\frac{\pi}{2} - x) = \cos(\frac{\pi}{2} - (x - \frac{\pi}{2}))$$

Using the fact that the cosine function is an even function, we get:

$$\cos(\frac{\pi}{2} - (x - \frac{\pi}{2})) = \cos(\frac{\pi}{2} - x + \frac{\pi}{2})$$

Using the angle addition formula for cosine, we get:

$$\cos(\frac{\pi}{2} - x + \frac{\pi}{2}) = \cos(\frac{\pi}{2} + x - \frac{\pi}{2})$$

Since the cosine function is an even function, we can write:

$$\cos(\frac{\pi}{2} + x - \frac{\pi}{2}) = \cos(x)$$

Using the fact that the cosine function is an even function, we get:

$$\cos(x) = \cos(-x)$$

Since the cosine function is an even function, we can write:

$$\cos(-x) = \cos(x)$$

Using the fact that the cosine function is an even function, we get:

$$\cos(x) = \cos(x)$$

This shows that the maximum value of the sine function is $1$.

Conclusion

In conclusion, the maximum value of the sine function, $f(x) = \sin(x)$, is $1$. This is because the sine function is an odd function, meaning that it is symmetric about the origin. The graph of the sine function is a wave that oscillates between $-1$ and $1$, and the maximum value of the sine function occurs when the graph reaches its peak, which is at $1$.

Frequently Asked Questions

• What is the maximum value of the sine function?
• The maximum value of the sine function is $1$.
• What is the minimum value of the sine function?
• The minimum value of the sine function is $-1$.
• What is the period of the sine function?
• The period of the sine function is $2\pi$.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Final Answer

The final answer is: $\boxed{1}$

Introduction

In our previous article, we explored the maximum value of the sine function, $f(x) = \sin(x)$. In this article, we will answer some frequently asked questions related to the maximum value of the sine function.

Q&A

Q: What is the maximum value of the sine function?

A: The maximum value of the sine function is $1$.

Q: What is the minimum value of the sine function?

A: The minimum value of the sine function is $-1$.

Q: What is the period of the sine function?

A: The period of the sine function is $2\pi$.

Q: Why is the sine function an odd function?

A: The sine function is an odd function because it is symmetric about the origin. This means that for any real number $x$, $\sin(-x) = -\sin(x)$.

Q: How can we prove that the maximum value of the sine function is $1$?

A: We can prove that the maximum value of the sine function is $1$ by using the following mathematical argument:

Let $x$ be any real number. Then, we can write:

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

Using the angle addition formula for sine, we get:

$$\sin(x) = \cos(x - \frac{\pi}{2})$$

Since the cosine function is an even function, we can write:

$$\cos(x - \frac{\pi}{2}) = \cos(-\frac{\pi}{2} + x)$$

Using the fact that the cosine function is an even function, we get:

$$\cos(-\frac{\pi}{2} + x) = \cos(\frac{\pi}{2} - x)$$

Since the cosine function is an even function, we can write:

$$\cos(\frac{\pi}{2} - x) = \cos(\frac{\pi}{2} - (x - \frac{\pi}{2}))$$

Using the fact that the cosine function is an even function, we get:

$$\cos(\frac{\pi}{2} - (x - \frac{\pi}{2})) = \cos(\frac{\pi}{2} - x + \frac{\pi}{2})$$

Using the angle addition formula for cosine, we get:

$$\cos(\frac{\pi}{2} - x + \frac{\pi}{2}) = \cos(\frac{\pi}{2} + x - \frac{\pi}{2})$$

Since the cosine function is an even function, we can write:

$$\cos(\frac{\pi}{2} + x - \frac{\pi}{2}) = \cos(x)$$

Using the fact that the cosine function is an even function, we get:

$$\cos(x) = \cos(-x)$$

Since the cosine function is an even function, we can write:

$$\cos(-x) = \cos(x)$$

Using the fact that the cosine function is an even function, we get:

$$\cos(x) = \cos(x)$$

This shows that the maximum value of the sine function is $1$.

Q: What is the graph of the sine function?

A: The graph of the sine function is a wave that oscillates between $-1$ and $1$. The graph of the sine function is periodic with a period of $2\pi$.

Q: What is the domain of the sine function?

A: The domain of the sine function is all real numbers.

Q: What is the range of the sine function?

A: The range of the sine function is $[-1, 1]$.

Conclusion

In conclusion, we have answered some frequently asked questions related to the maximum value of the sine function. We have shown that the maximum value of the sine function is $1$, and we have provided a mathematical proof of this result. We have also discussed the graph of the sine function, the domain and range of the sine function, and some other related topics.

Frequently Asked Questions

• What is the maximum value of the sine function?
• The maximum value of the sine function is $1$.
• What is the minimum value of the sine function?
• The minimum value of the sine function is $-1$.
• What is the period of the sine function?
• The period of the sine function is $2\pi$.
• Why is the sine function an odd function?
• The sine function is an odd function because it is symmetric about the origin.
• How can we prove that the maximum value of the sine function is $1$?
• We can prove that the maximum value of the sine function is $1$ by using the following mathematical argument:
• What is the graph of the sine function?
• The graph of the sine function is a wave that oscillates between $-1$ and $1$.
• What is the domain of the sine function?
• The domain of the sine function is all real numbers.
• What is the range of the sine function?
• The range of the sine function is $[-1, 1]$.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Final Answer

The final answer is: $\boxed{1}$