Questions: Rewrite cos(x - π/6) in terms of sin(x) and cos(x).

Solution Steps

To rewrite the cosine of a sum in terms of sine and cosine of the individual angles, apply the cosine addition formula, which expresses the cosine of a sum as the product of the cosines minus the product of the sines of the two angles.

We start with the expression \( \cos \left( x - \frac{\pi}{6} \right) \).

Using the cosine addition formula, we can rewrite \( \cos(a - b) \) as: \[ \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \] In our case, let \( a = x \) and \( b = \frac{\pi}{6} \). Thus, we have: \[ \cos \left( x - \frac{\pi}{6} \right) = \cos(x)\cos\left(\frac{\pi}{6}\right) + \sin(x)\sin\left(\frac{\pi}{6}\right) \]

We know that: \[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \quad \text{and} \quad \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \] Substituting these values into the expression gives: \[ \cos \left( x - \frac{\pi}{6} \right) = \cos(x) \cdot \frac{\sqrt{3}}{2} + \sin(x) \cdot \frac{1}{2} \]

This can be simplified to: \[ \cos \left( x - \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} \cos(x) + \frac{1}{2} \sin(x) \]

Final Answer