Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Find the exact value of the expression.
$\cos \frac{5 π}{12} \cos \frac{π}{12} + \sin \frac{5 π}{12} \sin \frac{π}{12}$

검증된 단계별 안내

Identify the given expression as the right side of the cosine difference formula: $\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$.

Match the angles in the expression to $\alpha$ and $\beta$. Here, $\alpha = \frac{5\pi}{12}$ and $\beta = \frac{\pi}{12}$.

Use the formula to rewrite the expression as $\cos\left(\frac{5\pi}{12} - \frac{\pi}{12}\right)$.

Simplify the angle inside the cosine: $\frac{5\pi}{12} - \frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3}$.

Recognize that the expression equals $\cos\left(\frac{\pi}{3}\right)$, and recall the exact value of $\cos\left(\frac{\pi}{3}\right)$ from the unit circle.

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주요 개념

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Cosine of a Difference Formula

The cosine of a difference between two angles α and β is given by cos(α - β) = cos α cos β + sin α sin β. This identity allows us to rewrite expressions involving sums of products of sines and cosines as a single cosine function, simplifying evaluation.

Exact Values of Trigonometric Functions at Special Angles

Certain angles, especially multiples of π/6, π/4, and π/3, have well-known exact sine and cosine values. Recognizing these angles helps in calculating exact trigonometric values without a calculator, which is essential for precise answers.

Angle Simplification and Periodicity

Trigonometric functions are periodic, so angles can be simplified by adding or subtracting multiples of 2π to find equivalent angles within a standard interval. This simplification aids in evaluating trigonometric expressions accurately.

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Find the exact value of the expression. cos5π12cosπ12+sin5π12sinπ12\(\cos\) \(\frac{5\pi}{12}\) \(\cos\) \(\frac{\pi}{12}\) + \(\sin\) \(\frac{5\pi}{12}\) \(\sin\) \(\frac{\pi}{12}\)