Using words, explain any difference in the solving and solutions for the equations: sin(2x) = 0.4 vs sin(x) = 0.4

Question

Don't need to actually solve, just list in words the differences

Answer 1

let's start thinking with:

Case One. x is an angle, if sinx equals a positive value (must be less than or equal to 1), then 0≤x≤π

and there are two angles 0≤x₁≤π/2 and π/2≤x₂≤π since x₂=π-x₁ Note x₁=x₂ when sinx=1

Case Two. x is an angle, if sin2x equals a positive value (must be less than or equal to 1), then 0≤2x₁≤π/2, or 0≤x₁≤π/4 and π/2≤2x₂≤π or π/4≤x₂≤π/2

think more, actually for case one, the addition of 2πk to x creates angles that have same value as sinx

k=0,+/-1,+/-2,...

so x₁+2πk and x₂+2πk are valid angles with same resulting value as sinx₁ and sinx₂

in case two., the addition of 2πk to 2x creates angles that have the same value as sin2x

k=0,+/-1,+/-2,...

so x₁+πk and x₂+πk are valid angles with same resulting value as sin2x₁ and sin2x₂

see sin[2(x₁+πk)]=sin2x₁

Answer 2

The second is the sine of angle x

The second is the sine fo twice the angle x