Trigonometric Equations on an Interval

Check angle units and the stated interval. Rearrange or factor without dividing away a possible zero factor. If the angle is transformed, solve over the transformed interval, then convert back.

Example

Solve \(2\sin^2x-\sin x-1=0\) for \(0\le x\le2\pi\).

\(\begin{aligned}(2\sin x+1)(\sin x-1)&=0\\\sin x&=-\dfrac12\quad\text{or}\quad\sin x=1\\x&=\dfrac{7\pi}{6},\ \dfrac{11\pi}{6},\ \dfrac\pi2.\end{aligned}\)

For \(\sin(2x)=\dfrac12\) on \(0\le x\le\pi\), solve on \(0\le2x\le2\pi\): \(2x=\dfrac\pi6,\dfrac{5\pi}6\), hence \(x=\dfrac\pi{12},\dfrac{5\pi}{12}\).

Graphically, intersect the two sides or find zeros of their difference. Use a window covering the entire interval, find every crossing or tangency, and check endpoints and undefined values in the original equation.

For a basic sine or cosine equation, first check that the requested value lies in \([-1,1]\). A calculator returns one principal angle, not a complete set. Locate the second angle using the unit circle, then include every turn lying in the stated interval. Tangent repeats after \(\pi\); sine and cosine repeat after \(2\pi\).

For \(u=bx+c\), calculate the images of both interval endpoints and put the bounds in increasing order. Keep each endpoint open or closed as specified. List all valid \(u\)-values before converting each by \(x=\dfrac{u-c}{b}\). A negative \(b\) reverses the interval orientation.

When a squared ratio occurs, retain both square roots: \(\tan^2u=k\), with \(k>0\), gives \(\tan u=\sqrt{k}\) or \(\tan u=-\sqrt{k}\). For a quadratic in sine or cosine, discard algebraic roots outside \([-1,1]\) before finding angles.

If an equation contains a product, solve each zero factor separately and merge repeated roots. Do not divide by \(\sin x\) or \(\cos x\) without checking its zero case. If tangent occurs, keep \(\cos x\ne0\) throughout; multiplying by a denominator does not make its excluded values valid.

A graphical root may be a crossing or a tangency. Use a window showing the entire interval and the relevant vertical range, then check suspicious points in the original equation. A jump across a vertical asymptote is not a root. When counting intersections, treat a shared endpoint only once.

Finish by substituting candidates into the original equation, checking the interval and angle mode, removing duplicates and sorting the answers. Keep exact multiples of \(\pi\) when available; otherwise round only the final values.