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Solving first degree trigonometric equations

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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.

What a first-degree trig equation is

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.

Solving a first-degree trig equation Solve 2 sin x minus 1 equals 0 for x from 0 to 360 degrees. Step 1: isolate sin x, giving sin x equals one half. Step 2: reference angle is 30 degrees. Step 3: sine is positive in quadrants 1 and 2, giving x equals 30 degrees and x equals 150 degrees. Solve: 2 sin x − 1 = 0, for 0° ≤ x < 360° Step 1Isolate: sin x = 1/2 Step 2Reference angle: 30° (since sin 30° = 1/2) Step 3Sine is positive in quadrants I and II Answerx = 30° and x = 150°
Isolate the trig ratio, find the reference angle, then use the quadrants where that ratio is positive (or negative) to list every solution.

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.

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