Using the Unit Circle to Find the Exact Value

Oct 13, 2025 by miodeya.com 49 views

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the fundamental concepts in trigonometry is the cosine function, which is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. In this article, we will explore how to find the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ .

Understanding the Problem

To find the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ , we need to first understand the concept of radians and the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. The angle $\theta$ is measured in radians, and the cosine of an angle is defined as the x-coordinate of the point on the unit circle that corresponds to the angle.

Using the Unit Circle to Find the Exact Value

We can use the unit circle to find the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ . To do this, we need to first find the point on the unit circle that corresponds to the angle $\frac{11 \pi}{8}$ . We can do this by using the fact that the angle $\frac{11 \pi}{8}$ is equivalent to the angle $\frac{3 \pi}{8}$ plus two full rotations of $2 \pi$ .

Finding the Reference Angle

The reference angle is the acute angle between the terminal side of the angle and the x-axis. In this case, the reference angle is $\frac{3 \pi}{8}$ . We can use the reference angle to find the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ .

Using the Reference Angle to Find the Exact Value

Using the reference angle $\frac{3 \pi}{8}$ , we can find the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ . To do this, we need to use the fact that the cosine of an angle is equal to the cosine of its reference angle.

Finding the Exact Value of $\cos \left(\frac{3 \pi}{8}\right)$

To find the exact value of $\cos \left(\frac{3 \pi}{8}\right)$ , we can use the half-angle formula for cosine. The half-angle formula for cosine is given by:

\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}

Applying the Half-Angle Formula

We can apply the half-angle formula to find the exact value of $\cos \left(\frac{3 \pi}{8}\right)$ . To do this, we need to first find the value of $\cos \left(\frac{\pi}{4}\right)$ .

Finding the Value of $\cos \left(\frac{\pi}{4}\right)$

The value of $\cos \left(\frac{\pi}{4}\right)$ is equal to $\frac{1}{\sqrt{2}}$ .

Applying the Half-Angle Formula Again

We can apply the half-angle formula again to find the exact value of $\cos \left(\frac{3 \pi}{8}\right)$ . To do this, we need to substitute the value of $\cos \left(\frac{\pi}{4}\right)$ into the half-angle formula.

Finding the Exact Value of $\cos \left(\frac{3 \pi}{8}\right)$

Using the half-angle formula, we can find the exact value of $\cos \left(\frac{3 \pi}{8}\right)$ .

\cos \left(\frac{3 \pi}{8}\right) = \pm \sqrt{\frac{1 + \frac{1}{\sqrt{2}}}{2}}

Simplifying the Expression

We can simplify the expression by rationalizing the denominator.

\cos \left(\frac{3 \pi}{8}\right) = \pm \sqrt{\frac{\sqrt{2} + 1}{2 \sqrt{2}}}

Rationalizing the Denominator

We can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$ .

\cos \left(\frac{3 \pi}{8}\right) = \pm \sqrt{\frac{\sqrt{2} + 1}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}}

Simplifying the Expression Again

We can simplify the expression again by canceling out the common factors.

\cos \left(\frac{3 \pi}{8}\right) = \pm \sqrt{\frac{2 + \sqrt{2}}{4}}

Finding the Exact Value of $\cos \left(\frac{11 \pi}{8}\right)$

The exact value of $\cos \left(\frac{11 \pi}{8}\right)$ is equal to the exact value of $\cos \left(\frac{3 \pi}{8}\right)$ .

\cos \left(\frac{11 \pi}{8}\right) = \pm \sqrt{\frac{2 + \sqrt{2}}{4}}

Conclusion

In this article, we explored how to find the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ . We used the unit circle and the reference angle to find the exact value of $\cos \left(\frac{3 \pi}{8}\right)$ , and then used the fact that the cosine of an angle is equal to the cosine of its reference angle to find the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ . The exact value of $\cos \left(\frac{11 \pi}{8}\right)$ is equal to $\pm \sqrt{\frac{2 + \sqrt{2}}{4}}$ .

Final Answer

The final answer is $\boxed{-\sqrt{\frac{2+\sqrt{2}}{4}}}$

Introduction

In our previous article, we explored how to find the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ . We used the unit circle and the reference angle to find the exact value of $\cos \left(\frac{3 \pi}{8}\right)$ , and then used the fact that the cosine of an angle is equal to the cosine of its reference angle to find the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ . In this article, we will answer some of the most frequently asked questions about finding the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ .

Q: What is the reference angle for $\frac{11 \pi}{8}$ ?

A: The reference angle for $\frac{11 \pi}{8}$ is $\frac{3 \pi}{8}$ .

Q: How do I find the exact value of $\cos \left(\frac{3 \pi}{8}\right)$ ?

A: To find the exact value of $\cos \left(\frac{3 \pi}{8}\right)$ , you can use the half-angle formula for cosine. The half-angle formula for cosine is given by:

\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}

Q: What is the value of $\cos \left(\frac{\pi}{4}\right)$ ?

A: The value of $\cos \left(\frac{\pi}{4}\right)$ is equal to $\frac{1}{\sqrt{2}}$ .

Q: How do I apply the half-angle formula to find the exact value of $\cos \left(\frac{3 \pi}{8}\right)$ ?

A: To apply the half-angle formula, you need to substitute the value of $\cos \left(\frac{\pi}{4}\right)$ into the formula.

Q: What is the exact value of $\cos \left(\frac{3 \pi}{8}\right)$ ?

A: The exact value of $\cos \left(\frac{3 \pi}{8}\right)$ is equal to $\pm \sqrt{\frac{2 + \sqrt{2}}{4}}$ .

Q: How do I find the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ ?

A: To find the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ , you can use the fact that the cosine of an angle is equal to the cosine of its reference angle.

Q: What is the final answer for $\cos \left(\frac{11 \pi}{8}\right)$ ?

A: The final answer for $\cos \left(\frac{11 \pi}{8}\right)$ is $\boxed{-\sqrt{\frac{2+\sqrt{2}}{4}}}$ .

Q: Why is the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ negative?

A: The exact value of $\cos \left(\frac{11 \pi}{8}\right)$ is negative because the angle $\frac{11 \pi}{8}$ is in the second quadrant, where the cosine function is negative.

Q: Can I use a calculator to find the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ ?

A: While you can use a calculator to find an approximate value of $\cos \left(\frac{11 \pi}{8}\right)$ , it is not recommended to use a calculator to find the exact value. Instead, you should use the unit circle and the reference angle to find the exact value.

Q: What are some common mistakes to avoid when finding the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ ?

A: Some common mistakes to avoid when finding the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ include:

Not using the unit circle and the reference angle to find the exact value
Not applying the half-angle formula correctly
Not rationalizing the denominator
Not simplifying the expression correctly

Conclusion

In this article, we answered some of the most frequently asked questions about finding the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ . We hope that this article has been helpful in clarifying any confusion and providing a better understanding of how to find the exact value of $\cos \left(\frac{11 \pi}{8}\right)$ .