Real periodic things rarely complete exactly one cycle every units. A tide model has , a voltage oscillates sixty times a second, and squeezes two whole cycles into the interval where fits one. So the equations you meet in applications almost always look like rather than . The good news is that you already know how to solve . Everything in this lesson is about one idea: treat the whole argument as a single angle, solve for it over the right interval, then convert back to . The one thing that can go wrong is the interval, and getting it right is what decides whether you find two solutions, four, or six.
Why a multiple angle multiplies the solutions
Let . As runs from to , the angle runs from to — two full trips around the unit circle. Each trip hits a value such as twice, so has four solutions in , not two. The method is therefore: substitute ; multiply the interval by ; find every in that widened interval (reference angle, CAST, then add until you leave the interval); finally divide every by . A fraction such as works the same way in reverse. Now runs only over , half a trip, so some angles you would normally list are simply not available, and the equation may have just one solution — or none.
| Equation on | Interval for the angle | Angles found | Solutions for |
|---|---|---|---|
| (six angles) | |||
| only ( is out of range) | |||
| none ( itself is excluded) | No solution |
Shifted angles such as 2x − π/3
The same substitution handles a phase shift. For , let and transform the interval exactly as you would transform : multiply by 2, then subtract . So becomes . Solve on that interval — — then undo the operations in reverse order: add , then divide by 2. Notice that the shifted interval starts at a negative angle, so it is worth checking whether a negative coterminal angle such as sneaks in. Here it does not, because .
When an identity has to come first
Substitution only works when ONE angle appears. If an equation mixes and , as in or , use a double-angle identity to rewrite everything in terms of , then factor exactly as in single-angle equations. Choose the version of that matches the other ratio present: next to , next to . Compound-angle formulas work in the other direction. An expression such as is the expanded form of ; recognizing it collapses two terms into a single sine of a shifted angle, which you then solve with the substitution method above. Whichever way you go, never divide both sides by a trig ratio, because the values that make it zero vanish from your answer.
Applied equations from sinusoidal models
A model such as for water depth, hours after midnight, turns every "when" question into a multiple-angle equation. Setting gives . Over one day, , the angle runs over , so expect four answers: , giving hours. The interval in an applied problem comes from the context — one day, one revolution, one year — so state it before solving, and interpret each answer with units: here the water is exactly m deep at 1 a.m., 5 a.m., 1 p.m. and 5 p.m.
