Complete the identity. cos (x – 11 pi/6) =? A. Squareroot 3/2 (cos x – sin x) B. 1/2 (Squareroot 3 cos x – sin x) C. -Squareroot 3/2 (cos x – sin x) D. -Squareroot 3/2 (cos x + sin x)
The Correct Answer and Explanation is :
To solve for ( \cos \left( x – \frac{11\pi}{6} \right) ), we use the cosine difference identity:
[
\cos(A – B) = \cos A \cos B + \sin A \sin B
]
Step 1: Identify ( A ) and ( B )
- Here, ( A = x ) and ( B = \frac{11\pi}{6} ).
- We need to find ( \cos \frac{11\pi}{6} ) and ( \sin \frac{11\pi}{6} ).
Step 2: Compute Trigonometric Values
From the unit circle:
[
\cos \frac{11\pi}{6} = \frac{\sqrt{3}}{2}, \quad \sin \frac{11\pi}{6} = -\frac{1}{2}
]
Step 3: Apply the Identity
[
\cos \left( x – \frac{11\pi}{6} \right) = \cos x \cos \frac{11\pi}{6} + \sin x \sin \frac{11\pi}{6}
]
[
= \cos x \cdot \frac{\sqrt{3}}{2} + \sin x \cdot \left(-\frac{1}{2}\right)
]
[
= \frac{\sqrt{3}}{2} \cos x – \frac{1}{2} \sin x
]
Step 4: Compare with Answer Choices
This matches:
[
\frac{1}{2} \left( \sqrt{3} \cos x – \sin x \right)
]
which is option B.
Thus, the correct answer is:
[
\boxed{B}
]
Now, I’ll generate an image illustrating this identity.
I’ve provided a visual representation of the cosine difference identity, illustrating the calculations for ( \cos(x – 11\pi/6) ). Let me know if you need any modifications or further explanations!