Solving First-Degree Trigonometric Equations

A first-degree trigonometric equation has the trig function appearing only to the first power, such as 2 sin x − 1 = 0. Solving it means isolating the trig function, then finding every angle in the given interval that satisfies it.

Step 1 — isolate the trig function using algebra, exactly as you would isolate x in a linear equation. Step 2 — find the reference angle, the acute angle whose trig ratio matches. Step 3 — use the ASTC rule (All, Sine, Tangent, Cosine positive in quadrants I, II, III, IV) to find every quadrant where the ratio has the right sign. Step 4 — write out each solution in the given interval.

Solve 2 sin x − 1 = 0 for 0° ≤ x < 360°. Isolating gives sin x = 1/2. The reference angle is 30°, since sin 30° = 1/2. Sine is positive in quadrants I and II, so the solutions are x = 30° and x = 180° − 30° = 150°.

Once you can solve first-degree equations, the same reference-angle method extends to second-degree trigonometric equations. Always check the interval given in the problem — some equations have infinitely many solutions if no interval is given, expressed with a "+ 360°n" term. Restricting to one interval, like 0° to 360°, keeps the answer to a short, specific list.

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• Trigonometry