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 2sin2x-sin x-1=0 for 0 <= x 2pi. (2sin x+1)(sin x-1)=0 sin x=- 12 x=1 x= 7pi6, 11pi6, 2. For sin(2x)= 12 on 0 <= x <= pi, solve on 0 2x 2pi: 2x= 6, 5pi6, hence x= 12, 5pi12. 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.

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.

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.

Trigonometry

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Trigonometric Equations on an Interval

Trigonometric Equations on an Interval