🎯Trigonometric Equations

Monk Tau keeps a single wheel of stone at the top of Radian Ridge, and he turns it slowly while he talks.

"Until now," he says, "you have walked the circle forwards. I give you an angle, you give me a ratio. sin 30° is 1/2, and there is nothing to argue about."

Then he stops the wheel.

"An equation walks the other way. I give you the ratio and ask you which angles produce it. And that question does not have one answer — it has as many as the circle chooses to give."

The reason is that the circle comes back. Two different points can sit at the same height, and two different points can sit at the same distance across. So sin t = 1/2 is true at 30° and again at 150°, and no amount of arithmetic will make one of those disappear.

Every problem in this skill is solved in the same two moves, in this order:

• ◆Size. Find the reference angle — the acute angle whose ratio has that size. Ignore the minus sign completely while you do it. sin t = -1/2 has reference angle 30°, because sin 30° = 1/2.
• ◆Sign. Decide which quadrants the answer lives in, and place the reference angle into each of them.

Sine is the y-coordinate, so it is positive above the x-axis: quadrants 1 and 2. Cosine is the x-coordinate, so it is positive to the right of the y-axis: quadrants 1 and 4. Tangent is y over x, so it is positive where the two signs agree: quadrants 1 and 3.

Placing the reference angle r is mechanical once you know the quadrant:

quadrant 1: t = r quadrant 2: t = 180° - r quadrant 3: t = 180° + r quadrant 4: t = 360° - r

So sin t = 1/2 gives 30° and 180° - 30° = 150°. And sin t = -1/2, with the same reference angle but the quadrants below the axis, gives 180° + 30° = 210° and 360° - 30° = 330°.

One warning about the tangent, because it does not behave like the other two. Tangent repeats every 180°, not every 360°. Its two answers in a full turn are always exactly half a turn apart: tan t = 1 gives 45° and 225°, not 45° and 315°.

Most equations do not arrive already saying "sin t = something". They arrive dressed up, and the dressing has to come off before the circle is any use at all.

The trick is to stop seeing sin t as trigonometry for a moment and see it as a single block. If you can solve 2x - 1 = 0 you can solve 2 sin t - 1 = 0, because it is the same three keystrokes:

2 sin t - 1 = 0 2 sin t = 1 sin t = 1/2 <- only now does the circle get involved t = 30°, 150°

Terms on both sides are no different — collect, then divide:

3 sin t + 2 = sin t + 1 2 sin t + 2 = 1 2 sin t = -1 sin t = -1/2 t = 210°, 330°

Two things go wrong here more than anything else, and both happen before the circle is ever reached.

The first is a dropped minus sign. 2 cos t + sqrt(3) = 0 leaves cos t = -sqrt(3)/2, not +sqrt(3)/2 — and that single character moves both answers from quadrants 1 and 4 to quadrants 2 and 3. Always write the isolated line out in full before you look anything up.

The second is stopping at one answer. Once you have the reference angle it is very tempting to write it down and move on. Ask yourself every time: which quadrants does this sign allow, and have I visited both of them?

There is one honest exception. A few ratios are reached at only one point of the whole circle, because they are as far as the function goes:

• ◆sin t = 1 only at 90°, and sin t = -1 only at 270°;
• ◆cos t = 1 only at 0°, and cos t = -1 only at 180°.

When a question lands on one of those it will tell you that there is one solution. Everywhere else, expect two.

When the same ratio appears twice, the equation is no longer linear, and the fix is the fix you already know from algebra: get everything on one side, factor, and set each factor to zero.

2 sin² t - sin t = 0 sin t (2 sin t - 1) = 0 sin t = 0 -> t = 0°, 180° 2 sin t - 1 = 0 -> sin t = 1/2 -> t = 30°, 150°

Four solutions: 0°, 30°, 150°, 180°. Notice that each factor was a fresh equation on the circle, and each one gave back two angles of its own. Two factors never means two answers.

Now the warning Tau repeats more than any other on the Ridge.

Never divide both sides by sin t or cos t.

It is so tempting. 2 sin² t = sin t looks as though one sin t should just cancel off each side, leaving 2 sin t = 1. But dividing by sin t is only legal when sin t is not zero — and the whole question is *which angles make things zero*. Divide, and 0° and 180° vanish from your answer without a sound. The equation is not wrong; it is simply missing half of itself.

The safe move is always the same: bring everything to one side and factor. Then nothing can be lost, because a product is zero exactly when one of its factors is zero.

There is one place where dividing IS safe, and it is worth knowing because it is the standard opening for a whole family of problems:

sin t = cos t tan t = 1 (divide both sides by cos t) t = 45°, 225°

Why is this one allowed? Because an angle with cos t = 0 would force sin t = 0 too for the equation to hold, and sine and cosine are never zero at the same angle — the point would have to sit at the origin, and every point on the unit circle is 1 away from it. So nothing can be lost. Any equation of the form a sin t = b cos t collapses to tan t = b/a this way.

A quadratic in disguise. When an equation has a squared ratio and a plain one, substitute. Let u = cos t, solve the ordinary quadratic, then put cos t back:

2 cos² t - 3 cos t + 1 = 0 2u² - 3u + 1 = 0 (2u - 1)(u - 1) = 0 cos t = 1/2 -> t = 60°, 300° cos t = 1 -> t = 0° three solutions: 0°, 60°, 300°

The two factors gave different numbers of angles, and that is normal. Solve each one on the circle and count what it actually returns.

When two different ratios appear. An equation such as 2 cos² t + sin t = 2 cannot be read anywhere, because the circle answers questions about one ratio at a time. Use the Pythagorean identity sin² t + cos² t = 1 to make every term speak the same language:

2 cos² t + sin t = 2 2(1 - sin² t) + sin t = 2 (cos² t = 1 - sin² t) 2 - 2 sin² t + sin t = 2 2 sin² t - sin t = 0 sin t (2 sin t - 1) = 0 t = 0°, 30°, 150°, 180°

Always convert towards whichever ratio appears unsquared, because that is the one you cannot rewrite.

A multiple angle. sin 2t = 1/2 is not sin t = 1/2 with a 2 attached, and the difference matters enormously. Rename the whole angle: let u = 2t. As t sweeps one turn, u sweeps two, so u runs over 0° <= u < 720°, and every base answer comes round twice:

sin u = 1/2, 0° <= u < 720° u = 30°, 150°, 390°, 510° (the first two, plus 360° each) t = u/2 = 15°, 75°, 195°, 255°

Four solutions, not two. In general, ratio(kt) = v has k times as many solutions as ratio(t) = v. The two habits that break it are forgetting the extra turns, and forgetting to divide by k at the end.

Halving works the same way in reverse, and catches people out because it goes the other direction: in tan(t/2) = 1, the angle u = t/2 only reaches 180°, so most of the circle is out of bounds and only u = 45° survives — a single answer, t = 90°.

Every solution, not just one turn. When no interval is given, describe the whole infinite family. Sine and cosine repeat every 360°, so each of their two answers needs a family of its own:

sin t = 1/2 -> t = 30° + 360°k or t = 150° + 360°k -> t = pi/6 + 2pi k or t = 5pi/6 + 2pi k

Tangent repeats every 180°, and its two answers are already half a turn apart, so one family covers both:

tan t = 1 -> t = 45° + 180°k -> t = pi/4 + pi k

Here k stands for any whole number, positive, negative or zero.

The answer box is narrower than the mathematics, so this skill has a small set of rules and every prompt tells you which one applies.

Two solutions on 0° <= t < 360°: type a pair, in degrees, smaller first.

sin t = 1/2 -> (30, 150) cos t = -1/2 -> (120, 240) tan t = 1 -> (45, 225)

Degrees only, and no ° sign is needed inside the brackets. Do not reorder them by size of reference angle or by quadrant — smaller number first, always.

One solution: type the whole number of degrees. cos t = -1 is typed as 180. The prompt says "there is one solution" whenever this is the case, so you are never left guessing whether a second answer is missing.

A solution set in radians is a choice, never a typed pair. The pair box reads numbers, so pi/6 cannot go in it. When an answer belongs in radians the question offers you four sets and you pick one:

{0, pi/6, 5pi/6, pi}

Read those left to right and check them against your own list before choosing. The commonest wrong option is the one that lost the solutions of a common factor — it will look neat and short, and it will be missing two angles.

A count is a whole number. "How many solutions does it have on 0° <= t < 360°?" is answered 4, not "four" and not a list.

A general solution is a choice too, because it is a sentence rather than a number. Check three things in the option you pick: the right first angle, the right period (180° for a tangent, 360° for a sine or a cosine), and the right number of families.

An application answers in whatever the question names. "At what angle does the seat first reach 17 m?" wants degrees; "how many seconds" wants seconds, and you get there by scaling — an angle of t degrees is t/360 of the full turn, so it takes t/360 of the period. Read the word FIRST carefully: two angles usually give the same height, and only the smaller one is being asked for.