Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Write the expression as the cosine of an angle.
$\cos \frac{5 π}{12} \cos \frac{π}{12} + \sin \frac{5 π}{12} \sin \frac{π}{12}$

검증된 단계별 안내

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

Compare the given expression $\cos \frac{5\pi}{12} \cos \frac{\pi}{12} + \sin \frac{5\pi}{12} \sin \frac{\pi}{12}$ to the formula and recognize that $\alpha = \frac{5\pi}{12}$ and $\beta = \frac{\pi}{12}$.

Apply the formula by rewriting the expression as $\cos \left( \frac{5\pi}{12} - \frac{\pi}{12} \right)$.

Simplify the angle inside the cosine function by subtracting the fractions: $\frac{5\pi}{12} - \frac{\pi}{12} = \frac{4\pi}{12}$.

Reduce the fraction $\frac{4\pi}{12}$ to its simplest form $\frac{\pi}{3}$, so the expression becomes $\cos \frac{\pi}{3}$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Cosine of a Difference Formula

The cosine of a difference of two angles, α and β, is given by cos(α - β) = cos α cos β + sin α sin β. This identity allows rewriting expressions involving products of sines and cosines as a single cosine function of the difference of angles.

Angle Simplification and Substitution

To rewrite the given expression as a single cosine, identify α and β from the terms. This involves recognizing the angles in radians and substituting them into the formula, then simplifying the difference α - β to find the equivalent angle.

Radian Measure and Angle Arithmetic

Understanding radian measure is essential for manipulating angles like 5π/12 and π/12. Adding or subtracting these fractions requires common denominators and simplification to find the exact angle represented by α - β.

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Write the expression as the cosine of an angle.cos5π12cosπ12+sin5π12sinπ12\(\cos\) \(\frac{5\pi}{12}\) \(\cos\) \(\frac{\pi}{12}\) + \(\sin\) \(\frac{5\pi}{12}\) \(\sin\) \(\frac{\pi}{12}\)