- Simplify the expression using \(\cos(-x) = \cos(x)\).
- Express \(\frac{\pi}{12}\) as \(\frac{\pi}{4} - \frac{\pi}{6}\) and apply the cosine difference formula.
- Calculate the exact value using standard trigonometric values: \(\frac{\sqrt{6} + \sqrt{2}}{4}\).
- Compare the result to the form \(\frac{\sqrt{A}(\sqrt{B} + 1)}{4}\) to find \(A=2\) and \(B=3\).
- Final Answer: \(\boxed{A=2, B=3}\).
Explanation
- Simplifying the Expression
We are asked to find the exact value of \(\cos \left(-\frac{\pi}{12}\right)\) and express it in the form \(\frac{\sqrt{A}(\sqrt{B} + 1)}{4}\). First, we use the parity property of the cosine function, which states that \(\cos(-x) = \cos(x)\). Therefore, \(\cos \left(-\frac{\pi}{12}\right) = \cos \left(\frac{\pi}{12}\right)\).
- Using the Difference Formula
To evaluate \(\cos \left(\frac{\pi}{12}\right)\), we can express \(\frac{\pi}{12}\) as a difference of standard angles whose cosine and sine values are known: \(\frac{\pi}{12} = \frac{\pi}{4} - \frac{\pi}{6}\). We then use the cosine difference formula, \(\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\), where \(\alpha = \frac{\pi}{4}\) and \(\beta = \frac{\pi}{6}\). Substituting these values, we get:
\[\cos \left(\frac{\pi}{4} - \frac{\pi}{6}\right) = \cos\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{6}\right) + \sin\left(\frac{\pi}{4}\right)\sin\left(\frac{\pi}{6}\right).\]
- Evaluating the Trigonometric Expression
Now we substitute the known trigonometric values:
- \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)
- \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\)
- \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)
- \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\)
Plugging these into our expression gives:
\[\left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}.\]
- Identifying the Constants
We want to match our result, \(\frac{\sqrt{6} + \sqrt{2}}{4}\), to the required form \(\frac{\sqrt{A}(\sqrt{B} + 1)}{4}\). We can factor out \(\sqrt{2}\) from the numerator:
\[\frac{\sqrt{6} + \sqrt{2}}{4} = \frac{\sqrt{2}(\sqrt{3} + 1)}{4}.\]
Comparing this to \(\frac{\sqrt{A}(\sqrt{B} + 1)}{4}\), we see that \(\sqrt{A} = \sqrt{2}\) and \(\sqrt{B} = \sqrt{3}\). Thus, we find \(A = 2\) and \(B = 3\). Therefore, the values are \(\boxed{A=2}\) and \(\boxed{B=3}\).
Examples
In architecture and engineering, precise calculation of angles is crucial when designing structures like arched roofs or curved glass panels. Knowing the exact value of trigonometric functions at specific angles, such as \(\cos(\frac{\pi}{12})\), allows engineers to calculate exact lengths and spans without rounding errors, ensuring the structural integrity of the design. For example, if a structural beam follows a curve defined by \(f(\theta) = L \cos(\theta)\), knowing the exact trigonometric values allows for precise fabrication of the beam's components.