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
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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.
More to read
9.5 Solving Trigonometric Equations
sin x = 0.5 has TWO solutions in 0 to 2π: π/6 and 5π/6, mirror images across π/2. The wave hits most heights twice per turn, so always hunt the twin. Reference angle first, then place both.
Sketch the wave or the circle to place them: first quadrant hit plus its mirror. Check each by substitution. One answer alone is half marks.
7.1 Solving Trigonometric Equations with Identities
Mixed equations must become one function first. For 2cos squared x = sin x + 1, replace cos squared with 1 − sin squared to get 2sin squared x + sin x − 1 = 0. Factorise like a quadratic in sin x.
Each factor gives its own solution set, so gather every angle in range. Never divide by a trig function, since that deletes solutions.
7.5 Solving Trigonometric Equations
For sin 2x = 0.5 with 0 ≤ x ≤ 2π, stretch FIRST: 2x runs 0 to 4π, covering two full turns and four hits. Solve for 2x, then halve each: π/12, 5π/12, 13π/12, 17π/12.
Forgetting the stretch loses half the answers. Write the stretched interval down before touching the equation. Interval first, solutions second, division last.
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17 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.