Evaluate The Expression: \sin \left(x-\frac{3 \pi}{4}\right ]

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Introduction

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In trigonometry, evaluating expressions involving sine and other trigonometric functions is a crucial aspect of problem-solving. The given expression, $\sin \left(x-\frac{3 \pi}{4}\right)$, involves the sine function and a phase shift. In this article, we will evaluate this expression and explore its properties.

Understanding the Sine Function

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The sine function is a fundamental concept in trigonometry, and it is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. The sine function is periodic, meaning that it repeats itself after a certain interval. The standard unit circle is used to represent the sine function, and it is defined as:

$$\sin(x) = \frac{y}{r}$$

where $x$ is the angle, $y$ is the $y$-coordinate of the point on the unit circle, and $r$ is the radius of the unit circle.

Evaluating the Expression

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To evaluate the expression $\sin \left(x-\frac{3 \pi}{4}\right)$, we need to use the angle subtraction formula for sine. The angle subtraction formula for sine states that:

$$\sin(a-b) = \sin(a)\cos(b) - \cos(a)\sin(b)$$

Using this formula, we can rewrite the expression as:

$$\sin \left(x-\frac{3 \pi}{4}\right) = \sin(x)\cos\left(\frac{3 \pi}{4}\right) - \cos(x)\sin\left(\frac{3 \pi}{4}\right)$$

Properties of the Sine Function

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The sine function has several important properties that are useful in evaluating expressions. Some of these properties include:

• Periodicity: The sine function is periodic, meaning that it repeats itself after a certain interval. The standard unit circle is used to represent the sine function, and it is defined as:

$$\sin(x) = \frac{y}{r}$$

• Symmetry: The sine function is symmetric about the origin, meaning that $\sin(-x) = -\sin(x)$.
• Range: The range of the sine function is $[-1, 1]$, meaning that the sine function can take on any value between $-1$ and $1$.

Evaluating the Expression Using the Angle Subtraction Formula

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Using the angle subtraction formula for sine, we can rewrite the expression as:

$$\sin \left(x-\frac{3 \pi}{4}\right) = \sin(x)\cos\left(\frac{3 \pi}{4}\right) - \cos(x)\sin\left(\frac{3 \pi}{4}\right)$$

To evaluate this expression, we need to find the values of $\sin(x)$ and $\cos(x)$. We can use the unit circle to find these values.

Finding the Values of $\sin(x)$ and $\cos(x)$

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Using the unit circle, we can find the values of $\sin(x)$ and $\cos(x)$ for any angle $x$. The unit circle is a circle with a radius of $1$ centered at the origin. The $x$-axis and $y$-axis are the coordinate axes.

Evaluating the Expression Using the Unit Circle

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Using the unit circle, we can evaluate the expression $\sin \left(x-\frac{3 \pi}{4}\right)$. We can find the values of $\sin(x)$ and $\cos(x)$ for any angle $x$.

Conclusion

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In conclusion, evaluating the expression $\sin \left(x-\frac{3 \pi}{4}\right)$ involves using the angle subtraction formula for sine and the unit circle. The angle subtraction formula for sine states that:

$$\sin(a-b) = \sin(a)\cos(b) - \cos(a)\sin(b)$$

Using this formula, we can rewrite the expression as:

$$\sin \left(x-\frac{3 \pi}{4}\right) = \sin(x)\cos\left(\frac{3 \pi}{4}\right) - \cos(x)\sin\left(\frac{3 \pi}{4}\right)$$

To evaluate this expression, we need to find the values of $\sin(x)$ and $\cos(x)$. We can use the unit circle to find these values.

Final Answer

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The final answer to the expression $\sin \left(x-\frac{3 \pi}{4}\right)$ is:

$$\sin \left(x-\frac{3 \pi}{4}\right) = \sin(x)\cos\left(\frac{3 \pi}{4}\right) - \cos(x)\sin\left(\frac{3 \pi}{4}\right)$$

This expression can be evaluated using the unit circle and the angle subtraction formula for sine.

References

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• Trigonometry: A comprehensive guide to trigonometry, including the sine function and the unit circle.
• Mathematics: A comprehensive guide to mathematics, including trigonometry and the sine function.

Future Work

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In the future, we can explore other trigonometric functions, such as the cosine function and the tangent function. We can also explore other mathematical concepts, such as calculus and differential equations.

Acknowledgments

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This article was written by [Your Name] and is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

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Introduction

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In our previous article, we evaluated the expression $\sin \left(x-\frac{3 \pi}{4}\right)$ using the angle subtraction formula for sine and the unit circle. In this article, we will answer some frequently asked questions (FAQs) related to this expression.

Q: What is the angle subtraction formula for sine?

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A: The angle subtraction formula for sine states that:

$$\sin(a-b) = \sin(a)\cos(b) - \cos(a)\sin(b)$$

Q: How do I evaluate the expression $\sin \left(x-\frac{3 \pi}{4}\right)$?

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A: To evaluate the expression $\sin \left(x-\frac{3 \pi}{4}\right)$, you need to use the angle subtraction formula for sine and the unit circle. You can find the values of $\sin(x)$ and $\cos(x)$ for any angle $x$ using the unit circle.

Q: What is the unit circle?

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A: The unit circle is a circle with a radius of $1$ centered at the origin. The $x$-axis and $y$-axis are the coordinate axes.

Q: How do I find the values of $\sin(x)$ and $\cos(x)$ using the unit circle?

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A: To find the values of $\sin(x)$ and $\cos(x)$ using the unit circle, you need to locate the point on the unit circle corresponding to the angle $x$. The $x$-coordinate of this point is $\cos(x)$, and the $y$-coordinate is $\sin(x)$.

Q: What is the range of the sine function?

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A: The range of the sine function is $[-1, 1]$, meaning that the sine function can take on any value between $-1$ and $1$.

Q: Is the sine function periodic?

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A: Yes, the sine function is periodic, meaning that it repeats itself after a certain interval. The standard unit circle is used to represent the sine function, and it is defined as:

$$\sin(x) = \frac{y}{r}$$

Q: What is the symmetry of the sine function?

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A: The sine function is symmetric about the origin, meaning that $\sin(-x) = -\sin(x)$.

Q: Can I use the angle addition formula for sine to evaluate the expression $\sin \left(x-\frac{3 \pi}{4}\right)$?

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A: No, you cannot use the angle addition formula for sine to evaluate the expression $\sin \left(x-\frac{3 \pi}{4}\right)$. The angle subtraction formula for sine is used to evaluate this expression.

Q: Can I use the unit circle to evaluate the expression $\sin \left(x-\frac{3 \pi}{4}\right)$?

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A: Yes, you can use the unit circle to evaluate the expression $\sin \left(x-\frac{3 \pi}{4}\right)$. You can find the values of $\sin(x)$ and $\cos(x)$ for any angle $x$ using the unit circle.

Q: What is the final answer to the expression $\sin \left(x-\frac{3 \pi}{4}\right)$?

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A: The final answer to the expression $\sin \left(x-\frac{3 \pi}{4}\right)$ is:

$$\sin \left(x-\frac{3 \pi}{4}\right) = \sin(x)\cos\left(\frac{3 \pi}{4}\right) - \cos(x)\sin\left(\frac{3 \pi}{4}\right)$$

Conclusion

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In conclusion, evaluating the expression $\sin \left(x-\frac{3 \pi}{4}\right)$ involves using the angle subtraction formula for sine and the unit circle. We have answered some frequently asked questions (FAQs) related to this expression, and we hope that this article has been helpful in understanding the concept of evaluating trigonometric expressions.

Final Answer

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The final answer to the expression $\sin \left(x-\frac{3 \pi}{4}\right)$ is:

$$\sin \left(x-\frac{3 \pi}{4}\right) = \sin(x)\cos\left(\frac{3 \pi}{4}\right) - \cos(x)\sin\left(\frac{3 \pi}{4}\right)$$

This expression can be evaluated using the unit circle and the angle subtraction formula for sine.

References

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• Trigonometry: A comprehensive guide to trigonometry, including the sine function and the unit circle.
• Mathematics: A comprehensive guide to mathematics, including trigonometry and the sine function.

Future Work

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In the future, we can explore other trigonometric functions, such as the cosine function and the tangent function. We can also explore other mathematical concepts, such as calculus and differential equations.

Acknowledgments

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This article was written by [Your Name] and is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.