A multiple-angle trig equation, like sin 2x equals one half, is solved by substituting u for the multiplied angle, solving over the correspondingly widened interval, and dividing the results by the multiplier. See a worked example showing why doubling the angle doubles the number of solutions.
Solving equations with a multiple angle
A multiple-angle trig equation, like sin 2x = ½, has the angle multiplied by a number (here, 2) instead of a plain x. Solving it takes one extra step compared to a single-angle equation: a substitution to keep track of the wider interval.
The substitution method
To solve sin 2x = ½ for 0° ≤ x < 360°, let u = 2x. Since x ranges up to 360°, u ranges up to 720° — twice as far, because doubling the angle means the full solution interval must double too.
Substituting u = 2x, solving for u over its doubled interval, then dividing by 2 to recover x.
Solving for u, then for x
Within 0° ≤ u < 720°, sin u = ½ has four solutions: u = 30°, 150°, 390°, and 510° (30° and 150° from the first cycle, plus another 360° added to each for the second cycle). Dividing each by 2 gives x = 15°, 75°, 195°, and 255°.
Why there are more solutions
A multiple-angle equation like sin 2x = ½ has twice as many solutions in the same interval as sin x = ½ would, because the angle 2x completes its cycle twice as fast. This builds on solving first-degree trig equations and uses the same reference angle ideas.