In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. b. Write the expression as the cosine of an angle.5π π 5π πcos ------- cos -------- + sin -------- sin -------12 12 12 12

Verified step by step guidance

1. Recognize the formula for the cosine of a difference: \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \).
2. Identify the given expression: \( \cos \frac{5\pi}{12} \cos \frac{\pi}{12} + \sin \frac{5\pi}{12} \sin \frac{\pi}{12} \).
3. Compare the given expression with the formula to identify \( \alpha \) and \( \beta \): \( \alpha = \frac{5\pi}{12} \) and \( \beta = \frac{\pi}{12} \).
4. Substitute \( \alpha \) and \( \beta \) into the formula: \( \cos(\alpha - \beta) = \cos\left(\frac{5\pi}{12} - \frac{\pi}{12}\right) \).
5. Simplify the expression inside the cosine: \( \cos\left(\frac{4\pi}{12}\right) = \cos\left(\frac{\pi}{3}\right) \).

Key Concepts

Cosine of Angle Difference Formula

The cosine of the difference of two angles, α and β, is given by the formula cos(α - β) = cos(α)cos(β) + sin(α)sin(β). This formula is essential for simplifying expressions involving the cosine of angle differences and is foundational in trigonometry.

Angle Measurement in Radians

In trigonometry, angles can be measured in degrees or radians. The question uses radians, where π radians equals 180 degrees. Understanding how to convert between these two systems is crucial for accurately interpreting and manipulating trigonometric expressions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. Familiarity with these identities, such as the Pythagorean identities and angle sum/difference identities, is vital for simplifying and solving trigonometric expressions effectively.