Use the following conditions to solve Exercises 1–4: 4 𝝅 sin α = ----- , ------- < α < 𝝅 5 2 5 𝝅 cos β = ------ , 0 < β < ------ 13 2 Find the exact value of each of the following. cos (α + β)

Sine and cosine are fundamental trigonometric functions that relate the angles of a right triangle to the ratios of its sides. The sine function, sin(θ), represents the ratio of the length of the opposite side to the hypotenuse, while the cosine function, cos(θ), represents the ratio of the adjacent side to the hypotenuse. Understanding these functions is crucial for solving problems involving angles and their relationships.

The angle addition formulas for sine and cosine allow us to find the sine or cosine of the sum of two angles. Specifically, cos(α + β) = cos(α)cos(β) - sin(α)sin(β). This formula is essential for calculating the cosine of the sum of angles when the individual angles are known, as in the given problem.

The range of angles specified in the problem indicates which quadrant the angles α and β lie in. For instance, the range for α is between 0 and π, placing it in the first or second quadrant, while β is between 0 and π/2, placing it in the first quadrant. Understanding the signs of sine and cosine in different quadrants is vital for determining the correct values of these functions when solving trigonometric equations.