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. Recognize that the expression cos(θ - 30°) is a cosine of a difference, which can be expanded using the cosine difference identity.
2. Recall the cosine difference identity: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).
3. Identify \( a = \theta \) and \( b = 30° \) in the identity.
4. Substitute \( a \) and \( b \) into the identity: \( \cos(\theta - 30°) = \cos \theta \cos 30° + \sin \theta \sin 30° \).
5. Use known values: \( \cos 30° = \frac{\sqrt{3}}{2} \) and \( \sin 30° = \frac{1}{2} \) to express the function in terms of \( \theta \) alone.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are essential for simplifying expressions and solving equations in trigonometry. Common identities include the Pythagorean identities, angle sum and difference identities, and reciprocal identities, which help in rewriting functions in terms of sine, cosine, and tangent.

Angle Sum and Difference Formulas

The angle sum and difference formulas allow us to express the sine, cosine, and tangent of the sum or difference of two angles in terms of the functions of those angles. For example, the cosine of the difference of two angles is given by cos(α - β) = cos(α)cos(β) + sin(α)sin(β). These formulas are crucial for rewriting expressions like cos(θ - 30°) in a more manageable form.

Function Notation and Variables

Understanding function notation and the role of variables is fundamental in trigonometry. In the expression cos(θ - 30°), θ represents an angle, and the expression itself is a function of θ. Recognizing how to manipulate these variables and their relationships is key to transforming trigonometric expressions and solving problems effectively.