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. Recognize that the expression \( \cos(90^\circ - \theta) \) involves the cosine of a difference.
2. Recall the co-function identity: \( \cos(90^\circ - \theta) = \sin(\theta) \).
3. Understand that this identity is derived from the complementary angle theorem, which states that the cosine of an angle is equal to the sine of its complement.
4. Apply the identity directly to rewrite \( \cos(90^\circ - \theta) \) as \( \sin(\theta) \).
5. Conclude that the expression \( \cos(90^\circ - \theta) \) simplifies to \( \sin(\theta) \), which is a trigonometric function of \( \theta \) alone.

Key Concepts

Cosine of a Sum or Difference Identity

The cosine of a sum or difference identity states that cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b). This identity allows us to express the cosine of an angle that is the sum or difference of two other angles in terms of the cosines and sines of those angles. It is essential for simplifying expressions involving trigonometric functions.

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. In trigonometry, this relationship leads to specific identities, such as cos(90° - θ) = sin(θ). Understanding complementary angles is crucial for transforming expressions involving angles that sum to 90 degrees into simpler trigonometric functions.

Trigonometric Function Transformation

Trigonometric function transformation involves rewriting trigonometric expressions using known identities to simplify or solve problems. This process often includes substituting one function for another, such as converting cos(90° - θ) into sin(θ). Mastery of these transformations is vital for effectively manipulating and solving trigonometric equations.