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

Verified step by step guidance

1. Recall the cosine difference identity: \(\cos(a - b) = \cos a \cos b + \sin a \sin b\).
2. Identify the angles in the expression: here, \(a = \theta\) and \(b = 30^\circ\).
3. Apply the identity to rewrite \(\cos(\theta - 30^\circ)\) as \(\cos \theta \cos 30^\circ + \sin \theta \sin 30^\circ\).
4. Substitute the exact values for \(\cos 30^\circ\) and \(\sin 30^\circ\): \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\sin 30^\circ = \frac{1}{2}\).
5. Write the final expression as \(\cos \theta \cdot \frac{\sqrt{3}}{2} + \sin \theta \cdot \frac{1}{2}\), which involves functions of \(\theta\) alone.

Key Concepts

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.