Every Solution of a Trigonometric Equation

Two answers in a turn, and then more

Solving sin x = ½ between 0° and 360° gives two answers. The inverse sine gives 30°, and the symmetry of the sine curve gives the second, 180° − 30° = 150°.

Without a range, those two are not the only answers. Adding a full turn of 360° brings a point on the unit circle back to the same place, so its sine cannot change. sin 390° = sin 30° = ½, and sin 510° = sin 150° = ½. A full turn the other way works too: sin(−330°) = ½.

The level line at a sine of 0.5 cuts the circle at two arms, 30° and 150°, and cuts the wave at those two angles in the first turn. Turn n to 1 and both solutions move on a full turn, to 390° and 510°. Turn it to −1 and they move back a full turn, to −330° and −210°.

One formula lists them all

Every solution is one of the first two plus a whole number of turns. Write n for that whole number, which can be positive, negative or zero. The general solution of sin x = ½ is x = 30° + 360°n or x = 150° + 360°n.

Test a member of each family. In the first, n = 1 gives 30° + 360° = 390°, and sin 390° = ½. In the second, n = −1 gives 150° − 360° = −210°, and sin(−210°) = ½, because −210° and 150° are the same position on the circle.

The same method works for any value the calculator gives. For sin x = 0.6, the inverse sine gives 36.87° to 2 decimal places, and the second angle is 180° − 36.87° = 143.13°. So x = 36.87° + 360°n or x = 143.13° + 360°n. With n = 1 these are 396.87° and 503.13°, and the sine of each is 0.6000 to 4 decimal places.

The two sine families in one line

The two families for a sine are often written together as $x=180^{\circ }n+{\left(-1\right)}^n\times 30^{\circ }$x = 180°n + (−1)ⁿ × 30°. When n is even, ${\left(-1\right)}^n$(−1)ⁿ is 1 and the formula adds 30°; when n is odd, ${\left(-1\right)}^n$(−1)ⁿ is −1 and it subtracts 30°.

Check the first four: n = 0 gives 30°, n = 1 gives 180° − 30° = 150°, n = 2 gives 360° + 30° = 390°, and n = 3 gives 540° − 30° = 510°. These are the same answers as the two families, taken in order.

Cosine: the two answers either side of 0°

For cos x = ½, the inverse cosine gives 60°. The cosine of an angle is its x-coordinate on the unit circle, and the point at −60° is the reflection of the point at 60° in the x-axis, so it has the same x-coordinate. So cos(−60°) = ½ as well.

Each of these repeats every full turn, so the general solution is x = 60° + 360°n or x = −60° + 360°n, written together as $x=\pm 60^{\circ }+360^{\circ }n$x = ±60° + 360°n. The second family includes 300°, the second answer between 0° and 360°: −60° + 360° = 300°.

The gold curve is y = cos x from −360° to 720°, with one square across for every 90°, and the plain straight line is y = ½. They meet at the dots: −300°, −60°, 60°, 300°, 420° and 660°. Each one is 60° or −60° plus a whole number of turns.

Tangent: one family, every half turn

For tan x = 1, the inverse tangent gives 45°. The tangent repeats every 180°, not every 360°, because half a turn takes the point on the circle to the opposite side, where both coordinates change sign and their ratio stays the same. So tan 225° = 1 as well.

Half a turn is enough to list every solution, so the tangent needs only one family: x = 45° + 180°n. With n = 1 it gives 225°, and with n = 2 it gives 405°, a full turn past 45°. A list that stepped by 360° would miss 225°.

The gold curve is y = tan x from −180° to 540°, with one square across for every 90°, and the plain straight line is y = 1. They meet once in each branch of the curve, at −135°, 45°, 225° and 405°, which are 180° apart.

In radians

A full turn is $2\pi$2π radians and a half turn is $\pi$π. So the general solution of sin x = ½ is $x=\frac{\pi }{6}+2n\pi$x = π/6 + 2nπ or $x=\frac{5\pi }{6}+2n\pi$x = 5π/6 + 2nπ, that of cos x = ½ is $x=\pm \frac{\pi }{3}+2n\pi$x = ±π/3 + 2nπ, and that of tan x = 1 is $x=\frac{\pi }{4}+n\pi$x = π/4 + nπ.

An interval is a filter on the list

A question that asks for the solutions in a range is answered from the general solution. Substitute n = −1, 0, 1, 2 in turn and keep the values that land in the range.

Solve sin x = ½ for $-360^{\circ }\le x\le 720^{\circ }$−360° ≤ x ≤ 720°. With n = −1 the families give −330° and −210°; with n = 0, 30° and 150°; with n = 1, 390° and 510°. With n = 2 they give 750° and 870°, which are too large, and with n = −2 they give −690° and −570°, which are too small. So there are six solutions: −330°, −210°, 30°, 150°, 390° and 510°.

The usual mistakes

Adding half a turn to a sine solution. 30° + 180° = 210°, but sin 210° = −½. The sine repeats every 360°, so the next solution past 360° is 30° + 360° = 390°.

Reflecting to 360° − 30° = 330° for a sine. That reflection is the cosine pairing, and sin 330° = −½, so 330° solves sin x = −½, not sin x = ½.

Stepping a tangent list by 360°. 45° + 360°n skips 225°, where the tangent is also 1.

Putting $\pm$± on a tangent list. tan(−45°) = −1, so $\pm 45^{\circ }+180^{\circ }n$±45° + 180°n includes angles that solve tan x = −1.

Stepping a cosine list by 180°. $\pm 60^{\circ }+180^{\circ }n$±60° + 180°n lands on 120° and 240°, where the cosine is −½.

Leaving out the second family. 60° + 360°n misses −60° and 300°, and 30° + 360°n misses 150°.

A weight passing a light beam

In the application below, the height of a weight on a spring is $8\sin \left(\frac{\pi x}{3}\right)$8 sin(πx/3) cm after x seconds. The general solution for the angle $\frac{\pi x}{3}$πx/3 is written first, with steps of $2\pi$2π, and only then multiplied by $\frac{3}{\pi }$3/π to give x. That turns the step of $2\pi$2π into a step of 6 seconds, the time for one bounce.