Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.
cos(90° - 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 \(A = 90^\circ\) and \(B = \theta\) in the expression \(\cos(90^\circ - \theta)\).
3. Apply the identity: \(\cos(90^\circ - \theta) = \cos 90^\circ \cos \theta + \sin 90^\circ \sin \theta\).
4. Use the known values: \(\cos 90^\circ = 0\) and \(\sin 90^\circ = 1\) to simplify the expression.
5. Simplify to get the expression as a trigonometric function of \(\theta\) alone.

Key Concepts

Cosine of a Difference Identity

The cosine of a difference identity states that cos(A - B) = cos A cos B + sin A sin B. This formula allows us to express the cosine of the difference between two angles in terms of the sines and cosines of the individual angles, which is essential for rewriting expressions like cos(90° - θ).

Special Angle Values in Trigonometry

Certain angles such as 0°, 30°, 45°, 60°, and 90° have known sine and cosine values. For example, cos 90° = 0 and sin 90° = 1. Using these values simplifies expressions involving these angles, enabling the reduction of cos(90° - θ) to a function involving only θ.

Co-Function Identities

Co-function identities relate trigonometric functions of complementary angles, such as cos(90° - θ) = sin θ. These identities are derived from the sum and difference formulas and are useful for expressing trigonometric functions in terms of a single variable.