- Identify the expression as the cosine subtraction identity: \(\cos(x - y) = \cos x \cos y + \sin x \sin y\).
- Simplify the angle difference \(\frac{3\pi}{7} - \frac{2\pi}{21} = \frac{\pi}{3}\) to find \(A = 3\).
- Evaluate \(\cos(\frac{\pi}{3}) = \frac{1}{2}\) to find \(B = 1\).
- Final results are \(A = \boxed{3}\) and \(B = \boxed{1}\).
Explanation
- Identify the relevant identity and variables
We are given the expression \(\cos \frac{3\pi}{7} \cos \frac{2\pi}{21} + \sin \frac{3\pi}{7} \sin \frac{2\pi}{21}\). To simplify this, we recall the trigonometric cosine subtraction identity:
\[\cos(x - y) = \cos x \cos y + \sin x \sin y.\]
By matching the given expression to this identity, we identify \(x = \frac{3\pi}{7}\) and \(y = \frac{2\pi}{21}\).
- Simplify the trigonometric expression
Using the identity, the original expression simplifies to \(\cos(x - y) = \cos\left(\frac{3\pi}{7} - \frac{2\pi}{21}\right)\).
To subtract these fractions, we find a common denominator, which is 21:
\[\frac{3\pi}{7} = \frac{3 \times 3\pi}{7 \times 3} = \frac{9\pi}{21}.\]
Now perform the subtraction:
\[\frac{9\pi}{21} - \frac{2\pi}{21} = \frac{7\pi}{21} = \frac{\pi}{3}.\]
Thus, the expression becomes \(\cos \frac{\pi}{3}\).
- Determine the values of A and B
We have found that \(\cos \frac{3\pi}{7} \cos \frac{2\pi}{21} + \sin \frac{3\pi}{7} \sin \frac{2\pi}{21} = \cos \frac{\pi}{3}\).
Comparing this to the given form \(\cos \frac{\pi}{A}\), we see that \(A = 3\).
Next, we evaluate \(\cos \frac{\pi}{3}\):
\[\cos \frac{\pi}{3} = \frac{1}{2}.\]
Comparing this to the form \(\frac{B}{2}\), we see that \(\frac{1}{2} = \frac{B}{2}\), which implies \(B = 1\).
Examples
Trigonometric identities like the cosine subtraction formula are foundational in engineering and physics, particularly in wave analysis and signal processing. For instance, when analyzing how multiple waves interact (superposition), using these identities allows engineers to simplify complex sums of oscillations into a single, manageable wave function, helping to determine the resulting amplitude and phase shift of combined signals.