Write each function as an expression involving functions of θ or x alone. See Example 2.
cos(45° - θ)

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, \(A = 45^\circ\) and \(B = \theta\).
3. Apply the identity to rewrite \(\cos(45^\circ - \theta)\) as \(\cos 45^\circ \cos \theta + \sin 45^\circ \sin \theta\).
4. Substitute the known exact values for \(\cos 45^\circ\) and \(\sin 45^\circ\), which are both \(\frac{\sqrt{2}}{2}\).
5. Write the final expression as \(\frac{\sqrt{2}}{2} \cos \theta + \frac{\sqrt{2}}{2} \sin \theta\), involving only functions of \(\theta\).

Key Concepts

Angle Difference Identity for Cosine

The angle difference identity states that cos(A - B) = cos A cos B + sin A sin B. This formula allows expressing the cosine of a difference of two angles as a combination of sines and cosines of the individual angles, facilitating simplification or evaluation.

Trigonometric Functions of Special Angles

Special angles like 45° have known exact sine and cosine values (e.g., cos 45° = sin 45° = √2/2). Using these values simplifies expressions involving these angles, making it easier to rewrite functions in terms of θ or x alone.

Function Notation and Variable Isolation

Rewriting trigonometric expressions to involve only one variable (θ or x) requires understanding function notation and how to isolate terms. This helps in expressing complex functions as simpler combinations of functions of a single variable.