Evaluate The Expression:$\sin \left(\frac{7 \pi}{12}+\frac{\pi}{12}\right$\]

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Introduction

In mathematics, trigonometric functions play a crucial role in solving various problems, especially in the field of calculus and algebra. One of the fundamental trigonometric functions is the sine function, which is used to describe the ratio of the length of the side opposite a given angle to the length of the hypotenuse in a right-angled triangle. In this article, we will evaluate the expression $\sin \left(\frac{7 \pi}{12}+\frac{\pi}{12}\right)$ using various trigonometric identities and properties.

Understanding the Expression

The given expression involves the sine function, which is a periodic function with a period of $2\pi$. This means that the value of the sine function repeats every $2\pi$ radians. The expression $\sin \left(\frac{7 \pi}{12}+\frac{\pi}{12}\right)$ can be simplified using the angle addition formula for sine, which states that $\sin (a+b) = \sin a \cos b + \cos a \sin b$.

Angle Addition Formula for Sine

The angle addition formula for sine is a fundamental identity in trigonometry, which allows us to simplify expressions involving the sum of two angles. Using this formula, we can rewrite the given expression as:

$$\sin \left(\frac{7 \pi}{12}+\frac{\pi}{12}\right) = \sin \frac{7 \pi}{12} \cos \frac{\pi}{12} + \cos \frac{7 \pi}{12} \sin \frac{\pi}{12}$$

Evaluating the Sine and Cosine Functions

To evaluate the expression, we need to find the values of $\sin \frac{7 \pi}{12}$, $\cos \frac{7 \pi}{12}$, $\sin \frac{\pi}{12}$, and $\cos \frac{\pi}{12}$. Using the unit circle or trigonometric identities, we can find these values as follows:

• $\sin \frac{7 \pi}{12} = \sin \left(\frac{\pi}{2} + \frac{\pi}{6}\right) = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$
• $\cos \frac{7 \pi}{12} = \cos \left(\frac{\pi}{2} + \frac{\pi}{6}\right) = -\sin \frac{\pi}{6} = -\frac{1}{2}$
• $\sin \frac{\pi}{12} = \sin \left(\frac{\pi}{3} - \frac{\pi}{4}\right) = \sin \frac{\pi}{3} \cos \frac{\pi}{4} - \cos \frac{\pi}{3} \sin \frac{\pi}{4} = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}$
• $\cos \frac{\pi}{12} = \cos \left(\frac{\pi}{3} - \frac{\pi}{4}\right) = \cos \frac{\pi}{3} \cos \frac{\pi}{4} + \sin \frac{\pi}{3} \sin \frac{\pi}{4} = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}$

Substituting the Values

Now that we have found the values of the sine and cosine functions, we can substitute them into the expression:

$$\sin \left(\frac{7 \pi}{12}+\frac{\pi}{12}\right) = \sin \frac{7 \pi}{12} \cos \frac{\pi}{12} + \cos \frac{7 \pi}{12} \sin \frac{\pi}{12}$$

$$= \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{6} + \sqrt{2}}{4} + \left(-\frac{1}{2}\right) \cdot \frac{\sqrt{6} - \sqrt{2}}{4}$$

Simplifying the Expression

To simplify the expression, we can combine the terms:

$$= \frac{\sqrt{3}(\sqrt{6} + \sqrt{2})}{8} - \frac{(\sqrt{6} - \sqrt{2})}{8}$$

$$= \frac{\sqrt{3}\sqrt{6} + \sqrt{3}\sqrt{2} - \sqrt{6} + \sqrt{2}}{8}$$

$$= \frac{\sqrt{18} + \sqrt{6} + \sqrt{2} - \sqrt{6} + \sqrt{2}}{8}$$

$$= \frac{\sqrt{18} + 2\sqrt{2}}{8}$$

$$= \frac{\sqrt{9 \cdot 2} + 2\sqrt{2}}{8}$$

$$= \frac{3\sqrt{2} + 2\sqrt{2}}{8}$$

$$= \frac{5\sqrt{2}}{8}$$

Conclusion

In this article, we evaluated the expression $\sin \left(\frac{7 \pi}{12}+\frac{\pi}{12}\right)$ using the angle addition formula for sine and the values of the sine and cosine functions. We found that the expression simplifies to $\frac{5\sqrt{2}}{8}$. This result demonstrates the importance of trigonometric identities and properties in solving mathematical problems.

Final Answer

The final answer is $\boxed{\frac{5\sqrt{2}}{8}}$.

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Introduction

In our previous article, we evaluated the expression $\sin \left(\frac{7 \pi}{12}+\frac{\pi}{12}\right)$ using the angle addition formula for sine and the values of the sine and cosine functions. In this article, we will answer some frequently asked questions related to this topic.

Q1: What is the angle addition formula for sine?

A1: The angle addition formula for sine is a fundamental identity in trigonometry, which states that $\sin (a+b) = \sin a \cos b + \cos a \sin b$.

Q2: How do I simplify the expression $\sin \left(\frac{7 \pi}{12}+\frac{\pi}{12}\right)$?

A2: To simplify the expression, you can use the angle addition formula for sine and the values of the sine and cosine functions. You can also use the unit circle or trigonometric identities to find the values of the sine and cosine functions.

Q3: What are the values of $\sin \frac{7 \pi}{12}$ and $\cos \frac{7 \pi}{12}$?

A3: The values of $\sin \frac{7 \pi}{12}$ and $\cos \frac{7 \pi}{12}$ are $\frac{\sqrt{3}}{2}$ and $-\frac{1}{2}$, respectively.

Q4: What are the values of $\sin \frac{\pi}{12}$ and $\cos \frac{\pi}{12}$?

A4: The values of $\sin \frac{\pi}{12}$ and $\cos \frac{\pi}{12}$ are $\frac{\sqrt{6} - \sqrt{2}}{4}$ and $\frac{\sqrt{6} + \sqrt{2}}{4}$, respectively.

Q5: How do I simplify the expression $\frac{\sqrt{3}\sqrt{6} + \sqrt{3}\sqrt{2} - \sqrt{6} + \sqrt{2}}{8}$?

A5: To simplify the expression, you can combine the terms and use the properties of radicals. You can also use the fact that $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$.

Q6: What is the final answer to the expression $\sin \left(\frac{7 \pi}{12}+\frac{\pi}{12}\right)$?

A6: The final answer to the expression $\sin \left(\frac{7 \pi}{12}+\frac{\pi}{12}\right)$ is $\boxed{\frac{5\sqrt{2}}{8}}$.

Q7: Why is the angle addition formula for sine important in trigonometry?

A7: The angle addition formula for sine is important in trigonometry because it allows us to simplify expressions involving the sum of two angles. This formula is used in a wide range of applications, including calculus, algebra, and geometry.

Q8: How do I use the unit circle to find the values of the sine and cosine functions?

A8: To use the unit circle to find the values of the sine and cosine functions, you can draw a diagram of the unit circle and identify the points corresponding to the given angles. You can then use the coordinates of these points to find the values of the sine and cosine functions.

Q9: What are some common applications of the angle addition formula for sine?

A9: Some common applications of the angle addition formula for sine include solving trigonometric equations, finding the values of the sine and cosine functions, and simplifying expressions involving the sum of two angles.

Q10: Why is it important to understand the properties of radicals in trigonometry?

A10: It is important to understand the properties of radicals in trigonometry because they are used to simplify expressions involving the square root of a number. This is particularly important in trigonometry, where the sine and cosine functions involve the square root of a number.

Conclusion

In this article, we answered some frequently asked questions related to the expression $\sin \left(\frac{7 \pi}{12}+\frac{\pi}{12}\right)$. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the topic.