I'm trying to solve equations like: sin 2x + sin x = 0 and cos 3x - cos x = 0 Should I use sum-to-product identities, or is it better to use double-angle formulas? What's the general strategy for finding all solutions in the interval [0, 2π)?

I'm studying abstract algebra and I understand the group axioms: 1. Closure: a · b ∈ G for all a, b ∈ G 2. Associativity: (a · b) · c = a · (b · c) 3. Identity: There exists e ∈ G such that e · a = a · e = a 4. Inverses: For each a, there exists a^(-1) such that a · a^(-1) = a^(-1) · a = e But I've heard that sometimes axioms 3 and 4 can be weakened. Specifically, if we have a semigroup (closure + associativity) with a left identity and left inverses, that is sufficient for a group. Could someone prove this? Also, what is the difference between a group, a monoid, and a semigroup?

I know how to compute determinants using cofactor expansion and row operations. But what does the determinant actually represent geometrically? For a 2 \ imes 2 matrix A = a & b c & d, det(A) = ad - bc. What does this number tell us about the transformation represented by A? And how does this generalise to 3 \ imes 3 and higher dimensions?

This is an IIT-JEE problem that has been bothering me: Find the number of solutions to the equation sin x = x/10 for x ∈ ℝ. I tried solving it analytically but it seems impossible due to the transcendental nature. My teacher suggested a graphical approach. Could someone explain how to count the intersections methodically? I know that both functions are odd, so the number of positive solutions will be symmetric to negative ones, with x = 0 always being a solution.

I am studying calculus and I need to compute the arc length of a curve defined parametrically. For a curve given by (x(t), y(t)) for a ≤ t ≤ b, the arc length formula is: L = ∫_a^b √(dx/dt\ ight)² + (dy/dt\ ight)² dt But where does this formula come from? And how do I apply it to a specific problem like finding the length of one arch of the cycloid x = t - sin t, y = 1 - cos t for 0 ≤ t ≤ 2π?