Use A Cosine Sum Or Difference Identity To Find The Exact Value Of Cos ⁡ 11 Π 12 \cos \frac{11 \pi}{12} Cos 12 11 Π ​ .

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Introduction

In trigonometry, the cosine function is a fundamental concept that plays a crucial role in solving various mathematical problems. The cosine function is used to measure the ratio of the adjacent side to the hypotenuse in a right-angled triangle. However, when dealing with angles in radians, it becomes challenging to find the exact value of the cosine function. In this article, we will explore how to use the cosine sum or difference identity to find the exact value of $\cos \frac{11 \pi}{12}$.

Understanding the Cosine Sum and Difference Identities

The cosine sum and difference identities are two fundamental formulas in trigonometry that help us find the exact value of the cosine function for various angles. The cosine sum identity states that:

$$\cos (A + B) = \cos A \cos B - \sin A \sin B$$

The cosine difference identity states that:

$$\cos (A - B) = \cos A \cos B + \sin A \sin B$$

These identities can be used to find the exact value of the cosine function for various angles by breaking down the angle into smaller components.

Breaking Down the Angle

To find the exact value of $\cos \frac{11 \pi}{12}$, we can break down the angle into smaller components using the cosine sum and difference identities. We can start by expressing $\frac{11 \pi}{12}$ as the sum of two angles: $\frac{5 \pi}{6}$ and $\frac{\pi}{12}$.

$$\frac{11 \pi}{12} = \frac{5 \pi}{6} + \frac{\pi}{12}$$

Applying the Cosine Sum Identity

Now that we have broken down the angle into smaller components, we can apply the cosine sum identity to find the exact value of $\cos \frac{11 \pi}{12}$.

$$\cos \left(\frac{5 \pi}{6} + \frac{\pi}{12}\right) = \cos \frac{5 \pi}{6} \cos \frac{\pi}{12} - \sin \frac{5 \pi}{6} \sin \frac{\pi}{12}$$

Finding the Exact Value of $\cos \frac{5 \pi}{6}$

To find the exact value of $\cos \frac{5 \pi}{6}$, we can use the fact that $\frac{5 \pi}{6}$ is in the second quadrant, where the cosine function is negative.

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

Using the cosine difference identity, we can rewrite the expression as:

$$\cos \frac{5 \pi}{6} = -\cos \frac{\pi}{6}$$

We know that $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$, so:

$$\cos \frac{5 \pi}{6} = -\frac{\sqrt{3}}{2}$$

Finding the Exact Value of $\cos \frac{\pi}{12}$

To find the exact value of $\cos \frac{\pi}{12}$, we can use the fact that $\frac{\pi}{12}$ is in the first quadrant, where the cosine function is positive.

$$\cos \frac{\pi}{12} = \cos \left(\frac{\pi}{3} - \frac{\pi}{4}\right)$$

Using the cosine difference identity, we can rewrite the expression as:

$$\cos \frac{\pi}{12} = \cos \frac{\pi}{3} \cos \frac{\pi}{4} + \sin \frac{\pi}{3} \sin \frac{\pi}{4}$$

We know that $\cos \frac{\pi}{3} = \frac{1}{2}$, $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$, and $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, so:

$$\cos \frac{\pi}{12} = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$$

Simplifying the expression, we get:

$$\cos \frac{\pi}{12} = \frac{\sqrt{6} + \sqrt{2}}{4}$$

Finding the Exact Value of $\sin \frac{5 \pi}{6}$

To find the exact value of $\sin \frac{5 \pi}{6}$, we can use the fact that $\frac{5 \pi}{6}$ is in the second quadrant, where the sine function is positive.

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

Using the sine difference identity, we can rewrite the expression as:

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

We know that $\sin \frac{\pi}{6} = \frac{1}{2}$, so:

$$\sin \frac{5 \pi}{6} = \frac{1}{2}$$

Finding the Exact Value of $\sin \frac{\pi}{12}$

To find the exact value of $\sin \frac{\pi}{12}$, we can use the fact that $\frac{\pi}{12}$ is in the first quadrant, where the sine function is positive.

$$\sin \frac{\pi}{12} = \sin \left(\frac{\pi}{3} - \frac{\pi}{4}\right)$$

Using the sine difference identity, we can rewrite the expression as:

$$\sin \frac{\pi}{12} = \sin \frac{\pi}{3} \cos \frac{\pi}{4} - \cos \frac{\pi}{3} \sin \frac{\pi}{4}$$

We know that $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$, $\cos \frac{\pi}{3} = \frac{1}{2}$, $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, and $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, so:

$$\sin \frac{\pi}{12} = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$$

Simplifying the expression, we get:

$$\sin \frac{\pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}$$

Substituting the Values

Now that we have found the exact values of $\cos \frac{5 \pi}{6}$, $\cos \frac{\pi}{12}$, $\sin \frac{5 \pi}{6}$, and $\sin \frac{\pi}{12}$, we can substitute these values into the expression for $\cos \frac{11 \pi}{12}$.

$$\cos \frac{11 \pi}{12} = \cos \left(\frac{5 \pi}{6} + \frac{\pi}{12}\right) = \cos \frac{5 \pi}{6} \cos \frac{\pi}{12} - \sin \frac{5 \pi}{6} \sin \frac{\pi}{12}$$

Substituting the values, we get:

$$\cos \frac{11 \pi}{12} = \left(-\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) - \left(\frac{1}{2}\right) \left(\frac{\sqrt{6} - \sqrt{2}}{4}\right)$$

Simplifying the expression, we get:

$$\cos \frac{11 \pi}{12} = -\frac{\sqrt{18} + \sqrt{6}}{8} - \frac{\sqrt{6} - \sqrt{2}}{8}$$

Combining like terms, we get:

$$\cos \frac{11 \pi}{12} = -\frac{\sqrt{18} + \sqrt{6} + \sqrt{6} - \sqrt{2}}{8}$$

Simplifying the expression further, we get:

$$\cos \frac{11 \pi}{12} = -\frac{\sqrt{18} + 2\sqrt{6} - \sqrt{2}}{8}$$

Conclusion

In this article, we used the cosine sum and difference identities to find the exact value of $\cos \frac{11 \pi}{12}$. We broke down the angle into smaller components and applied the cosine sum identity to find the exact value of the cosine function. We also used the fact that $\frac{5 \pi}{6}$ is in the second quadrant, where the cosine function is negative, and that $\frac{\pi}{12}$ is in the first quadrant, where the cosine function is positive. We found the exact values of $\cos \frac

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Introduction

In our previous article, we used the cosine sum and difference identities to find the exact value of $\cos \frac{11 \pi}{12}$. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the cosine sum identity?

A: The cosine sum identity is a formula that helps us find the exact value of the cosine function for various angles. It states that:

$$\cos (A + B) = \cos A \cos B - \sin A \sin B$$

Q: What is the cosine difference identity?

A: The cosine difference identity is a formula that helps us find the exact value of the cosine function for various angles. It states that:

$$\cos (A - B) = \cos A \cos B + \sin A \sin B$$

Q: How do I use the cosine sum and difference identities to find the exact value of $\cos \frac{11 \pi}{12}$?

A: To find the exact value of $\cos \frac{11 \pi}{12}$, we can break down the angle into smaller components using the cosine sum and difference identities. We can start by expressing $\frac{11 \pi}{12}$ as the sum of two angles: $\frac{5 \pi}{6}$ and $\frac{\pi}{12}$.

$$\frac{11 \pi}{12} = \frac{5 \pi}{6} + \frac{\pi}{12}$$

Q: How do I find the exact value of $\cos \frac{5 \pi}{6}$?

A: To find the exact value of $\cos \frac{5 \pi}{6}$, we can use the fact that $\frac{5 \pi}{6}$ is in the second quadrant, where the cosine function is negative.

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

Using the cosine difference identity, we can rewrite the expression as:

$$\cos \frac{5 \pi}{6} = -\cos \frac{\pi}{6}$$

We know that $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$, so:

$$\cos \frac{5 \pi}{6} = -\frac{\sqrt{3}}{2}$$

Q: How do I find the exact value of $\cos \frac{\pi}{12}$?

A: To find the exact value of $\cos \frac{\pi}{12}$, we can use the fact that $\frac{\pi}{12}$ is in the first quadrant, where the cosine function is positive.

$$\cos \frac{\pi}{12} = \cos \left(\frac{\pi}{3} - \frac{\pi}{4}\right)$$

Using the cosine difference identity, we can rewrite the expression as:

$$\cos \frac{\pi}{12} = \cos \frac{\pi}{3} \cos \frac{\pi}{4} + \sin \frac{\pi}{3} \sin \frac{\pi}{4}$$

We know that $\cos \frac{\pi}{3} = \frac{1}{2}$, $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$, and $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, so:

$$\cos \frac{\pi}{12} = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$$

Simplifying the expression, we get:

$$\cos \frac{\pi}{12} = \frac{\sqrt{6} + \sqrt{2}}{4}$$

Q: How do I find the exact value of $\sin \frac{5 \pi}{6}$?

A: To find the exact value of $\sin \frac{5 \pi}{6}$, we can use the fact that $\frac{5 \pi}{6}$ is in the second quadrant, where the sine function is positive.

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

Using the sine difference identity, we can rewrite the expression as:

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

We know that $\sin \frac{\pi}{6} = \frac{1}{2}$, so:

$$\sin \frac{5 \pi}{6} = \frac{1}{2}$$

Q: How do I find the exact value of $\sin \frac{\pi}{12}$?

A: To find the exact value of $\sin \frac{\pi}{12}$, we can use the fact that $\frac{\pi}{12}$ is in the first quadrant, where the sine function is positive.

$$\sin \frac{\pi}{12} = \sin \left(\frac{\pi}{3} - \frac{\pi}{4}\right)$$

Using the sine difference identity, we can rewrite the expression as:

$$\sin \frac{\pi}{12} = \sin \frac{\pi}{3} \cos \frac{\pi}{4} - \cos \frac{\pi}{3} \sin \frac{\pi}{4}$$

We know that $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$, $\cos \frac{\pi}{3} = \frac{1}{2}$, $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, and $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, so:

$$\sin \frac{\pi}{12} = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$$

Simplifying the expression, we get:

$$\sin \frac{\pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}$$

Q: How do I substitute the values into the expression for $\cos \frac{11 \pi}{12}$?

A: To substitute the values into the expression for $\cos \frac{11 \pi}{12}$, we can use the cosine sum identity.

$$\cos \frac{11 \pi}{12} = \cos \left(\frac{5 \pi}{6} + \frac{\pi}{12}\right) = \cos \frac{5 \pi}{6} \cos \frac{\pi}{12} - \sin \frac{5 \pi}{6} \sin \frac{\pi}{12}$$

Substituting the values, we get:

$$\cos \frac{11 \pi}{12} = \left(-\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) - \left(\frac{1}{2}\right) \left(\frac{\sqrt{6} - \sqrt{2}}{4}\right)$$

Simplifying the expression, we get:

$$\cos \frac{11 \pi}{12} = -\frac{\sqrt{18} + \sqrt{6}}{8} - \frac{\sqrt{6} - \sqrt{2}}{8}$$

Combining like terms, we get:

$$\cos \frac{11 \pi}{12} = -\frac{\sqrt{18} + 2\sqrt{6} - \sqrt{2}}{8}$$

Conclusion

In this article, we answered some frequently asked questions related to finding the exact value of $\cos \frac{11 \pi}{12}$ using the cosine sum and difference identities. We provided step-by-step solutions to each question and explained the concepts in detail. We hope that this article has been helpful in understanding the cosine sum and difference identities and how to use them to find the exact value of the cosine function for various angles.