Write each function as an expression involving functions of θ or x alone. See Example 2.
cos(θ - 30°)

검증된 단계별 안내

Recall the cosine difference identity: \(\cos(a - b) = \cos a \cos b + \sin a \sin b\).

Identify the angles in the expression: here, \(a = \theta\) and \(b = 30^\circ\).

Apply the identity to rewrite \(\cos(\theta - 30^\circ)\) as \(\cos \theta \cos 30^\circ + \sin \theta \sin 30^\circ\).

Substitute the exact values for \(\cos 30^\circ\) and \(\sin 30^\circ\): \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\sin 30^\circ = \frac{1}{2}\).

Write the final expression as \(\cos \theta \cdot \frac{\sqrt{3}}{2} + \sin \theta \cdot \frac{1}{2}\), which involves functions of \(\theta\) alone.

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주요 개념

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

Angle Difference Identity for Cosine

The angle difference identity states that cos(α - β) = cos α cos β + sin α sin β. This formula allows expressing the cosine of a difference of two angles as a combination of sines and cosines of the individual angles, which is essential for rewriting cos(θ - 30°) in terms of cos θ and sin θ.

Trigonometric Functions of Special Angles

Special angles like 30°, 45°, and 60° have known exact sine and cosine values. For 30°, cos 30° = √3/2 and sin 30° = 1/2. Using these values simplifies expressions involving these angles, enabling the conversion of cos(θ - 30°) into a function involving θ alone.

Function Notation and Variable Dependence

Understanding that trigonometric functions can be expressed as functions of a single variable (θ or x) is crucial. This means rewriting expressions so that all terms depend only on θ, facilitating further manipulation or evaluation without ambiguity about the variables involved.

Write each function as an expression involving functions of θ or x alone. See Example 2.cos(θ - 30°)