Find the exact value of each expression. (Do not use a calculator.)
cos (7π/9) cos (2π/9) - sin (7π/9) sin (2π/9)

검증된 단계별 안내

Recognize that the expression matches the cosine addition formula: \(\cos A \cos B - \sin A \sin B = \cos(A + B)\).

Identify the angles in the expression: \(A = \frac{7\pi}{9}\) and \(B = \frac{2\pi}{9}\).

Apply the formula by substituting the values of \(A\) and \(B\): \(\cos\left(\frac{7\pi}{9}\right) \cos\left(\frac{2\pi}{9}\right) - \sin\left(\frac{7\pi}{9}\right) \sin\left(\frac{2\pi}{9}\right) = \cos\left(\frac{7\pi}{9} + \frac{2\pi}{9}\right)\).

Add the angles inside the cosine function: \(\frac{7\pi}{9} + \frac{2\pi}{9} = \frac{9\pi}{9} = \pi\).

Use the known exact value of \(\cos \pi\) to find the final exact value.

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주요 개념

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

Cosine of Sum and Difference Identities

These identities express the cosine of a sum or difference of two angles in terms of the sines and cosines of the individual angles. Specifically, cos(A + B) = cos A cos B - sin A sin B, which matches the given expression and allows simplification without a calculator.

Exact Values of Trigonometric Functions at Special Angles

Certain angles, especially those related to π/3, π/4, π/6, and their multiples, have known exact sine and cosine values. Recognizing or converting angles to these special angles helps in finding exact trigonometric values without approximation.

Angle Measurement in Radians

Radians measure angles based on the radius of a circle, where 2π radians equal 360 degrees. Understanding radian measure is essential for interpreting and manipulating trigonometric expressions involving π, such as 7π/9 and 2π/9.

Find the exact value of each expression. (Do not use a calculator.)cos (7π/9) cos (2π/9) - sin (7π/9) sin (2π/9)