Solving Basic Equations

One Answer Is Never Enough

Solve $\sin x = \frac{1}{2}$.

Your first thought is $x = 30°$, and that is right. But so is $150°$. And $390°$. And $510°$. The answers never stop.

The dashed line is the value we want. It crosses the wave again and again, and every crossing is a solution. Because sine repeats forever, so do the answers.

The Two Jobs

Solving a trig equation splits into two tasks:

1. Find the solutions inside one cycle
2. Add the period to capture all the repeats

Get those two right and you have every answer.

Step 1: The Reference Angle

Start by ignoring signs. What acute angle has a sine of $\frac{1}{2}$?

$\arcsin\tfrac{1}{2} = 30°$

This $30°$ is the reference angle, the building block for the real solutions.

Step 2: Pick the Quadrants

The sign of the value tells you which quadrants the solutions live in.

Sine is positive in the first and second quadrants, so:

• First quadrant: $x = 30°$
• Second quadrant: $x = 180° - 30° = 150°$

Those are the two solutions in one full turn. In the animation, they are the green and yellow dots of the first cycle.

Step 3: Add the Period

Sine repeats every $360°$, so each solution spawns an infinite family:

$x = 30° + 360°k \qquad x = 150° + 360°k$

for any integer $k$. That covers all of them at once. This is called the general solution.

Cosine and Tangent Are a Little Different

The idea is the same, but the pattern of solutions changes with each function.

Tangent stands out: it repeats every $180°$, not $360°$, so it needs only one family.

Example: A Cosine Equation

Solve $\cos x = -\frac{1}{2}$.

The reference angle comes from the positive value: $\arccos\frac{1}{2} = 60°$.

Cosine is negative in the second and third quadrants:

• $x = 180° - 60° = 120°$
• $x = 180° + 60° = 240°$

So the general solution is:

$x = 120° + 360°k \qquad x = 240° + 360°k$

Example: A Tangent Equation

Solve $\tan x = 1$.

The reference angle is $\arctan 1 = 45°$. Since tangent repeats every $180°$, one family says it all:

$x = 45° + 180°k$