Trigonometric Equations

A trigonometric equation like sin x = k asks where the wave y = sin x crosses a horizontal line at height k. The x-coordinates of those crossings are the solutions, and moving the line changes how many appear in one period: two, one, or none. Equations use a reference angle and the matching quadrants, while inequalities become the interval where the curve lies above or below the line. Drag the line y = k and watch the number of intersections change.

Intersections Are the Solutions

Horizontal line y = k0.5

👀 See It

①The solutions of sin x = k are the x-coordinates where y=sin x meets y=k

②Raising or lowering k changes the count per period to 0, 1, or 2

③An inequality is the interval where the curve is above (or below) the line

Reference Angle and Quadrant

Solving trig equations

sin x = k ⇒ find the reference angle and collect solutions by quadrant

On [0,2π), sin·cos usually give 2; boundary values (±1, 0) give 1

tan equations

tan x = k ⇒ one solution every period π

Since tan has period π, there are 2 solutions on [0,2π)

Inequalities Become Intervals

📐 Read Above vs Below

①sin x > k is the x-interval where the curve is above the line y=k

②First solve sin x = k (the boundaries)

③Between the boundaries, check above/below to fix the interval

Solve It Directly

Example 1

Find all solutions of sin x = 1/2 on 0 ≤ x < 2π.

The reference angle is π/6 (sin(π/6)=1/2).

Collect solutions in quadrants 1 and 2 where sin is positive.

x = π/6, 5π/6

▸ x = π/6, 5π/6

Fix the reference angle first, then pick only the quadrants matching the sign.

Example 2

Solve cos x < 1/2 on 0 ≤ x < 2π.

The boundary cos x = 1/2 has solutions x = π/3, 5π/3.

Between them, cos is less than 1/2.

π/3 < x < 5π/3

▸ π/3 < x < 5π/3

For inequalities, solve the boundary equation first, then check the sign between.

Wrap-up

Core strategy

Equation: reference angle + quadrant / Inequality: boundary then interval

Sketching the solution positions on the graph and unit circle first is safest

2022 KICE mock exam Math type, adapted

On 0 ≤ x < 2π, what is the sum of all solutions of 2 sin x = √3?

①π/2

②2π/3

③π

④4π/3

⑤3π/2

▸ ③ π

sin x = √3/2 with reference angle π/3.

The quadrant-1,2 solutions are x = π/3, 2π/3, so the sum is

π/3 + 2π/3 = π

🎯 Exam Points

①Visualize solutions as graph/unit-circle intersections

②sin·cos usually give 2 per period, tan has period π

③For inequalities, solve the boundary first → judge the interval

④Always check the domain like [0,2π)

⑤Memorize special-angle values √3/2, 1/2, √2/2

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Trig Applications

Arithmetic Sequence

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