Solving Trigonometric Equations in a Given Interval
Find every solution inside a stated interval, using the graph or the circle to get the solutions the calculator does not show, and adjusting the interval first when the angle is 2x or x + 30°.
What a learner can do afterwards
- Solve sin x = 0.5 for 0 ≤ x ≤ 2π and give both solutions
- Stretch the interval before solving an equation written in 2x
- Use an identity to reduce an equation to a single trigonometric function first
1 · Read
Most heights are hit twice per turn. Sin x equals 0.5 gives pi over 6 and 5 pi over 6, mirror images across pi over 2. Find the reference angle first, then place both. One answer alone is half marks.
For sin 2x equals 0.5 with x between 0 and 2 pi, stretch first. Then 2x runs from 0 to 4 pi, covering two full turns. Solve for 2x, then halve each hit: pi over 12, 5 pi over 12, 13 pi over 12, 17 pi over 12.
Mixed equations must become one function first. Replace cos squared with 1 minus sin squared. The result factors like a quadratic in sin x. Each factor gives its own angles, so gather every one in range.
Never divide by a trig function, since that deletes solutions. Check each answer by substitution. Sketch the wave or the circle to confirm both twins are placed.
Place both twins, stretch the interval first, reduce to one function, and check each answer.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.