Which Expression Is Equivalent To Sin ⁡ 7 Π 6 \sin \frac{7 \pi}{6} Sin 6 7 Π ​ ?A. Sin ⁡ Π 6 \sin \frac{\pi}{6} Sin 6 Π ​ B. Sin ⁡ 5 Π 6 \sin \frac{5 \pi}{6} Sin 6 5 Π ​ C. Sin ⁡ 5 Π 3 \sin \frac{5 \pi}{3} Sin 3 5 Π ​ D. Sin ⁡ 11 Π 6 \sin \frac{11 \pi}{6} Sin 6 11 Π ​

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on solving trigonometric expressions, specifically the expression $\sin \frac{7 \pi}{6}$.

Understanding the Problem

The problem asks us to find an equivalent expression for $\sin \frac{7 \pi}{6}$. To solve this, we need to understand the properties of the sine function and how it behaves with different angles.

The Sine Function

The sine function is a periodic function that oscillates between -1 and 1. 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 has a period of $2 \pi$, which means that it repeats itself every $2 \pi$ radians.

Properties of the Sine Function

The sine function has several important properties that we need to know to solve trigonometric expressions. These properties include:

• Periodicity: The sine function has a period of $2 \pi$, which means that $\sin (x + 2 \pi) = \sin x$.
• Symmetry: The sine function is an odd function, which means that $\sin (-x) = -\sin x$.
• Co-function identities: The sine function has several co-function identities, including $\sin \left( \frac{\pi}{2} - x \right) = \cos x$ and $\sin \left( \frac{\pi}{2} + x \right) = \cos \left( \frac{\pi}{2} - x \right)$.

Solving the Expression

Now that we have a good understanding of the sine function and its properties, let's solve the expression $\sin \frac{7 \pi}{6}$.

We can start by using the periodicity property of the sine function. Since the sine function has a period of $2 \pi$, we can subtract $2 \pi$ from the angle to get an equivalent expression.

$\sin \frac{7 \pi}{6} = \sin \left( \frac{7 \pi}{6} - 2 \pi \right)$

Simplifying the expression, we get:

$\sin \frac{7 \pi}{6} = \sin \left( \frac{7 \pi}{6} - \frac{12 \pi}{6} \right)$

$\sin \frac{7 \pi}{6} = \sin \left( \frac{-5 \pi}{6} \right)$

Using the symmetry property of the sine function, we can rewrite the expression as:

$\sin \frac{7 \pi}{6} = -\sin \frac{5 \pi}{6}$

Now, we can use the co-function identity $\sin \left( \frac{\pi}{2} - x \right) = \cos x$ to rewrite the expression as:

$\sin \frac{7 \pi}{6} = -\cos \frac{\pi}{6}$

Evaluating the Expression

Now that we have simplified the expression, let's evaluate it.

We know that $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$. Therefore, we can substitute this value into the expression:

$\sin \frac{7 \pi}{6} = -\frac{\sqrt{3}}{2}$

Conclusion

In this article, we solved the expression $\sin \frac{7 \pi}{6}$ using the properties of the sine function. We used the periodicity, symmetry, and co-function identities of the sine function to simplify the expression and evaluate it.

Answer

The final answer is:

• $\sin \frac{7 \pi}{6}$ is equivalent to $\boxed{-\frac{\sqrt{3}}{2}}$

Comparison of Options

Now that we have solved the expression, let's compare our answer with the options provided:

• A. $\sin \frac{\pi}{6}$: This is not equivalent to $\sin \frac{7 \pi}{6}$.
• B. $\sin \frac{5 \pi}{6}$: This is not equivalent to $\sin \frac{7 \pi}{6}$.
• C. $\sin \frac{5 \pi}{3}$: This is not equivalent to $\sin \frac{7 \pi}{6}$.
• D. $\sin \frac{11 \pi}{6}$: This is equivalent to $\sin \frac{7 \pi}{6}$.

Therefore, the correct answer is:

• D. $\sin \frac{11 \pi}{6}$

Final Thoughts

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Q: What is the period of the sine function?

A: The period of the sine function is $2 \pi$, which means that it repeats itself every $2 \pi$ radians.

Q: What is the symmetry property of the sine function?

A: The sine function is an odd function, which means that $\sin (-x) = -\sin x$.

Q: What is the co-function identity of the sine function?

A: The co-function identity of the sine function is $\sin \left( \frac{\pi}{2} - x \right) = \cos x$.

Q: How do I simplify a trigonometric expression?

A: To simplify a trigonometric expression, you can use the following steps:

1. Use the periodicity property to subtract $2 \pi$ from the angle.
2. Use the symmetry property to rewrite the expression as a negative angle.
3. Use the co-function identity to rewrite the expression in terms of the cosine function.

Q: How do I evaluate a trigonometric expression?

A: To evaluate a trigonometric expression, you can use the following steps:

1. Simplify the expression using the periodicity, symmetry, and co-function identities.
2. Use the values of the sine and cosine functions for common angles to evaluate the expression.

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

A: The sine and cosine functions are both trigonometric functions that describe the relationships between the sides and angles of triangles. However, the sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse, while the cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Q: How do I use the unit circle to evaluate trigonometric expressions?

A: The unit circle is a circle with a radius of 1 that is centered at the origin of the coordinate plane. To use the unit circle to evaluate trigonometric expressions, you can use the following steps:

1. Draw a point on the unit circle that corresponds to the angle you want to evaluate.
2. Use the coordinates of the point to evaluate the sine and cosine functions.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• $\sin^2 x + \cos^2 x = 1$
• $\sin (x + y) = \sin x \cos y + \cos x \sin y$
• $\cos (x + y) = \cos x \cos y - \sin x \sin y$

Q: How do I use trigonometric identities to simplify expressions?

A: To use trigonometric identities to simplify expressions, you can use the following steps:

1. Identify the trigonometric identity that corresponds to the expression you want to simplify.
2. Use the identity to rewrite the expression in a simpler form.

Q: What are some common applications of trigonometry?

A: Some common applications of trigonometry include:

• Navigation: Trigonometry is used to calculate distances and directions between locations.
• Physics: Trigonometry is used to describe the motion of objects and the relationships between forces and motion.
• Engineering: Trigonometry is used to design and build structures such as bridges and buildings.

Q: How do I practice trigonometry?

A: To practice trigonometry, you can use the following steps:

1. Practice solving trigonometric expressions using the periodicity, symmetry, and co-function identities.
2. Practice evaluating trigonometric expressions using the unit circle.
3. Practice using trigonometric identities to simplify expressions.
4. Practice applying trigonometry to real-world problems.