Find complete interval and general solutions

Try

Find every angle in one full turn whose unit-circle height is 1/2. Draw the horizontal line y=1/2 through the circle. It meets the circle at π/6 and 5π/6. Both points are required; an arcsine calculation returns only the first principal angle.

Predict

Predict the full set if angles are allowed to include any number of revolutions. Each of the two branches repeats after 2π, so the solutions are x=π/6+2kπ or x=5π/6+2kπ, with k any integer. Negative integers are necessary to include negative rotations.

Experiment

Solve sin x(2cos x−1)=0 on 0≤x<2π. Separate the zero-product branches before dividing anything.

Branch	Condition	Solutions in the interval
sin x=0	Circle height zero	0, π
cos x=1/2	Circle horizontal coordinate 1/2	π/3, 5π/3

Plot all four points on the angle axis. Dividing the original equation by sin x would discard the first row. Combining branch lists means taking their union and removing duplicate values, not adding their multiplicities.

Understand

A reliable solution sequence is: record the original domain, simplify using valid identities, solve every algebraic branch, restore periodic repetitions, then filter to the requested interval. Check candidates in the original equation if any step could introduce or remove values.

For tan(3x)=1, first solve the angle equation 3x=π/4+kπ. Divide the entire expression by 3 to obtain x=π/12+kπ/3. Dividing only the principal angle and forgetting to divide the period shift gives an incomplete family.

For sin(2x)=√3/2, the angle 2x has two families: π/3+2kπ and 2π/3+2kπ. Therefore x=π/6+kπ or π/3+kπ. The resulting period in x is π, reflecting the doubled input frequency.

Endpoints matter. On [0,2π], sin x=0 includes 0, π and 2π. On [0,2π), the last endpoint is excluded. Although 0 and 2π name the same terminal point, they are distinct real inputs when both belong to the requested interval.

A quotient equals zero only when its numerator is zero and its denominator is defined and nonzero. For sin x/cos x=0, the candidates from sine must pass the cosine check. Squaring or using an identity in a larger domain can similarly introduce candidates that the original expression forbids.

Apply

Use written general-solution notation for irrational angle families; the numeric checker is not a symbolic angle parser. Explain the integer parameter and why the branch list is complete. A graph can help count solutions on a bounded interval, but it should corroborate the branch reasoning rather than replace it.

Continue

Periodic equations become useful in modeling heights, signals and recurring measurements. The next lesson fits a sinusoid and tests what the observations actually establish.