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Solving First-Degree Trigonometric Equations
A first-degree trigonometric equation has the trig function to the first power, such as 2 sine x minus 1 equals 0. Solve by isolating the trig ratio, finding the reference angle, using the quadrants where that ratio has the correct sign, and listing every solution in the given interval, with a worked example.
The four-step method
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.
Worked example
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°.
Checking the domain
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.