https://file.aiquickdraw.com/mathbot/file/1726326595761vducz4dc.jpg

We are asked to find the exact value of \(\cos\left(\frac{13\pi}{12}\right)\) by using a sum or difference formula. The angle \(\frac{13\pi}{12}\) is not a standard angle, so we need to express it as a sum or difference of two angles that are standard angles for which we know the cosine values.

We can break down \(\frac{13\pi}{12}\) as:

\[
\frac{13\pi}{12} = \pi - \frac{\pi}{4}
\]

This works because \(\pi = \frac{12\pi}{12}\), and \(\frac{13\pi}{12} = \frac{12\pi}{12} + \frac{\pi}{12}\), making the expression equivalent to \(\pi - \frac{\pi}{4}\).

Now, using the **cosine difference identity**:
\[
\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)
\]

Let \(A = \pi\) and \(B = \frac{\pi}{4}\). Substituting these into the formula:

\[
\cos\left(\frac{13\pi}{12}\right) = \cos(\pi)\cos\left(\frac{\pi}{4}\right) + \sin(\pi)\sin\left(\frac{\pi}{4}\right)
\]

From the unit circle, we know:
- \(\cos(\pi) = -1\)
- \(\sin(\pi) = 0\)
- \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)
- \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)

Now substitute these values:

\[
\cos\left(\frac{13\pi}{12}\right) = (-1) \cdot \frac{\sqrt{2}}{2} + 0 \cdot \frac{\sqrt{2}}{2}
\]

Simplifying this gives:

\[
\cos\left(\frac{13\pi}{12}\right) = -\frac{\sqrt{2}}{2}
\]

Thus, the exact value of \(\cos\left(\frac{13\pi}{12}\right)\) is:

\[
\boxed{-\frac{\sqrt{2}}{2}}
\]

Would you like a detailed explanation on how to use this in similar problems?

### Related questions:
1. How do you apply the sum and difference identities for sine?
2. What is the unit circle, and how does it help in trigonometry?
3. Can \(\cos\left(\frac{13\pi}{12}\right)\) be calculated without a sum or difference identity?
4. How do you derive the cosine and sine values for standard angles like \(\frac{\pi}{4}\)?
5. What other angle sum or difference combinations could work for \(\frac{13\pi}{12}\)?

### Tip:
Always ensure you are working with angles that are easily recognizable from the unit circle to simplify trigonometric calculations!

Find the exact value of cos(13π/12) by using a sum or difference formula.

Learn how to find the exact value of cos(13π/12) by applying the cosine difference identity. This problem explores trigonometric identities, unit circle values, and the application of the cosine difference formula, suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation