The equation looks for inputs

A trigonometric equation asks for the values ​​of the variable that make an equality true. Before transforming the expression, determine its domain. Divisions by functions that can be zero can eliminate solutions.

For $\sin x=m$ or $\cos x=m$, $-1\le m\le1$ is required. For the tangent, any real $m$ is possible, but entries where $\cos x=0$ remain prohibited.

Demonstration of sine families

A horizontal line cuts the cycle at a maximum of two points. If one of them corresponds to $\alpha$, the other corresponds to $\pi-\alpha$. As full turns repeat the stitch:

$$\sin x=\sin\alpha\iff \begin{cases} x=\alpha+2k\pi,\ \text{ou}\\ x=\pi-\alpha+2k\pi, \end{cases}\qquad k\in\mathbb Z.$$

At extreme values ​​of the sine, the two families can coincide; this does not add distinct solutions.

Cosine and tangent

A vertical line gives the positions of $\alpha$ and $-\alpha$:

$$\cos x=\cos\alpha\iff x=\pm\alpha+2k\pi.$$

Two opposite directions have the same vertical/horizontal ratio:

$$\tan x=\tan\alpha\iff x=\alpha+k\pi,$$

as long as $\tan\alpha$ exists. These formulas also resolve equalities between expression functions: replace $x$ and $\alpha$ with the corresponding arguments and solve the resulting equations.

Complete example

Solve $\sin2x=\sin(\pi/3)$. The possibilities are:

$$2x=\frac\pi3+2k\pi\Rightarrow x=\frac\pi6+k\pi,$$

$$2x=\pi-\frac\pi3+2k\pi\Rightarrow x=\frac\pi3+k\pi.$$

Both families are needed. The equality $\sin u=\sin v$ does not just imply $u=v$.

Reference cases

$$\sin x=0\iff x=k\pi,$$

$$\cos x=0\iff x=\frac\pi2+k\pi,$$

$$\tan x=0\iff x=k\pi.$$

In degrees, change $\pi$ to $180°$ and $2\pi$ to $360°$, keeping the same unit in all terms.

In everyday life

  • Ferris wheel. Finding the moments when the cabin reaches a certain height is solving $\operatorname{sen}\theta = m$.
  • Outlet. The voltage of 60 Hz cancels out 120 times per second: the solutions of a fundamental equation.
  • Tides. High tide times form a family of solutions, one for each period.
The cabin heights: each target height gives two solutions per turn.

Exercises with resolution

1. Solve $\cos x=-1/2$ in all reals.

Check resolution 1

$x=2\pi/3+2k\pi$ or $x=4\pi/3+2k\pi$, $k\in\mathbb Z$.

2. Solve for $\tan(2x)=1$.

Check resolution 2

$2x=\pi/4+k\pi$, then $x=\pi/8+k\pi/2$. In these entries the tangent of the argument is defined.

In the interactive book

The same subjects, with all the demonstrations and the complete list of exercises, are in the interactive Trigonometry book:

This class is a shorter introduction to the topic; In the book, the theory is accompanied by solved and proposed exercises from the chapter.

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