Rewrite cos(x-7pi/6) in terms of sin(x) and cos(x)

To rewrite \(\cos(x - \frac{7\pi}{6})\) in terms of \(\sin(x)\) and \(\cos(x)\), we can use the angle subtraction formula for cosine:

\[
\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)
\]

Here, \(a = x\) and \(b = \frac{7\pi}{6}\). So,

\[
\cos\left(x - \frac{7\pi}{6}\right) = \cos(x)\cos\left(\frac{7\pi}{6}\right) + \sin(x)\sin\left(\frac{7\pi}{6}\right)
\]

Next, we need to determine \(\cos\left(\frac{7\pi}{6}\right)\) and \(\sin\left(\frac{7\pi}{6}\right)\):

\[
\frac{7\pi}{6} = \pi + \frac{\pi}{6}
\]

Since \(\frac{7\pi}{6}\) is in the third quadrant, where cosine is negative and sine is negative, we have:

\[
\cos\left(\frac{7\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}
\]

\[
\sin\left(\frac{7\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2}
\]

Substitute these values into the formula:

\[
\cos\left(x - \frac{7\pi}{6}\right) = \cos(x)\left(-\frac{\sqrt{3}}{2}\right) + \sin(x)\left(-\frac{1}{2}\right)
\]

Simplify:

\[
\cos\left(x - \frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}\cos(x) - \frac{1}{2}\sin(x)
\]

Therefore, \(\cos\left(x - \frac{7\pi}{6}\right)\) in terms of \(\sin(x)\) and \(\cos(x)\) is:

\[
\cos\left(x - \frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}\cos(x) - \frac{1}{2}\sin(x)
\]

Would you like more details or have any questions?

Here are some related questions:

1. How can you rewrite \(\sin(x - \frac{7\pi}{6})\) in terms of \(\sin(x)\) and \(\cos(x)\)?
2. What is the value of \(\cos\left(\frac{7\pi}{6}\right)\)?
3. What is the value of \(\sin\left(\frac{7\pi}{6}\right)\)?
4. How can you use the angle addition formula for cosine?
5. Can you show how to derive the angle subtraction formula for cosine?
6. What is the cosine of a sum or difference of angles in the third quadrant?
7. How does the quadrant affect the signs of sine and cosine?
8. How can trigonometric identities help simplify expressions involving angles?

**Tip:** Always remember to consider the quadrant of the angle when determining the signs of sine and cosine values.

Learn how to rewrite cos(x - 7π/6) in terms of sin(x) and cos(x) using the angle subtraction formula for cosine. Understand the trigonometric identities involved and their application. Suitable for high school students and anyone studying trigonometry.

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