(Originais Teachy 2024) - Question Easy of Mathematics

In a physics experiment, a block of mass 'm' is placed on an inclined plane with an angle of elevation 'θ' relative to the horizontal. To analyze the acting forces, it is convenient to decompose the weight of the block 'm' into two components: one parallel to the inclined plane and another perpendicular to it. The parallel component, denoted by 'P_//', is given by P_// = m * g * sin(θ), where 'g' is the acceleration due to gravity. The perpendicular component, denoted by 'P_⊥', is given by P_⊥ = m * g * cos(θ). To simplify the calculations, it is common to rationalize the sin(θ) and cos(θ) expressions that appear in the equations. Step 1: We start with P_// = m * g * sin(θ) and P_⊥ = m * g * cos(θ). Step 2: We rationalize the denominator of sin(θ) by taking a step that results in a new equation P_//' = m * g * a / b (where 'a' and 'b' are numbers) and keep P_⊥ as the original equation. Step 3: After rationalizing, we verify that P_//' = P_//. Based on this context and the indicated steps, answer the following questions: (1) Develop the reasoning of step 2 and show how to rationalize the denominator of sin(θ) to obtain the expression P_//'. (2) After rationalizing, prove that the obtained expression P_//' is equal to the original expression P_//, demonstrating that sin(θ) / √n = √n / k, where 'n' and 'k' are rational numbers.