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### What is a trigonometric equation?

A trigonometric equation is any equation that contains a trigonometric function otherwise from trigonometric identities that are true for any angle a trigonometric equation may be true only for some certain angles in some cases.

### How to solve the equation $\displaystyle \sin x=a$?

Step 1: We find an angle $\displaystyle \alpha$ that its sinuses is equal to $\displaystyle a$. (if an angle like thisexists)

Step 2: Another angle that has the sinuses to $\displaystyle a$is the angle $\displaystyle 18{{0}^{\circ }}-\alpha$.

Step 3: All the solutions of the equation $\displaystyle \sin x=a$ are:

$\displaystyle x=k\cdot {{360}^{\circ }}+\alpha$

$\displaystyle x=k\cdot {{360}^{\circ }}+18{{0}^{\circ }}-\alpha$   where $\displaystyle k\in Z$.

Using $\displaystyle \pi$ instead of degrees the solutions can be written like:

$\displaystyle x=2k\pi +\alpha$

$\displaystyle x=2k\pi +\pi -\alpha$ where $\displaystyle k\in Z$

Tip! To find the value of $\displaystyle \alpha$ we can use the calculator,  of the four basic functions or from the graph of the trigonometric function.

Example 1: Solve the equation $\displaystyle \sin x=\frac{1}{2}$.

Step 1: An angle that is known to have the sinuses $\displaystyle \frac{1}{2}$ is $\displaystyle \alpha =3{{0}^{\circ }}$.

Step 2: Another angle that has the same sinuses is $\displaystyle 18{{0}^{\circ }}-3{{0}^{\circ }}=150$.

Step 3: All the angles that has the sinuses $\displaystyle \frac{1}{2}$ are:

$\displaystyle x=k\cdot {{360}^{\circ }}+3{{0}^{\circ }}$ and $\displaystyle x=k\cdot {{360}^{\circ }}+{{150}^{\circ }}$, or $\displaystyle x=2k\pi +\frac{\pi }{6}$ and $\displaystyle x=2k\pi +\frac{{5\pi }}{6}$, where $\displaystyle k\in Z$

Note! By substituting k with different integers we get the solutions of our equation above:

$\displaystyle k=1$ then $\displaystyle \left\{ {{{{390}}^{\circ }}{{{,510}}^{\circ }}} \right\}$

$\displaystyle k=-1$ then $\displaystyle \left\{ {-{{{330}}^{\circ }},-{{{210}}^{\circ }}} \right\}$

Example 2: Solve the equation $\displaystyle \sin 3x=\frac{1}{2}$.

$\displaystyle \alpha =3{{0}^{\circ }}$

$\displaystyle 3x=k\cdot {{360}^{\circ }}+3{{0}^{\circ }}$

$\displaystyle x=k\cdot {{120}^{\circ }}+{{10}^{\circ }}$ and $\displaystyle 3x=k\cdot {{360}^{\circ }}+{{150}^{\circ }}$

$\displaystyle x=k\cdot {{120}^{{}^\circ }}+{{50}^{{}^\circ }},~~k\in Z$

So the solutions of the equations are: $\displaystyle x=k\cdot {{120}^{\circ }}+{{10}^{\circ }}$

$\displaystyle x=k\cdot {{120}^{{}^\circ }}+{{50}^{{}^\circ }}$$\displaystyle k\in Z$

How to solve the equation $\displaystyle \cos x=b$?

Step 1: We find an angle $\displaystyle \alpha$ that its cosine is equal to $\displaystyle b$ (if an angle like this exists)

Step 2: Another angle that has the cosine to $\displaystyle b$ is the angle $\displaystyle -\alpha$

Step 3: All the solutions of the equation $\displaystyle \cos x=b$are:

$\displaystyle x=k\cdot {{360}^{\circ }}\pm \alpha$ or $\displaystyle x=2k\pi \pm \alpha$, $\displaystyle k\in Z$

Example 3: Solve the equation $\displaystyle \cos x=-\frac{1}{2}$.

Step 1: An angle that is known to have the cosine $\displaystyle -\frac{1}{2}$is $\displaystyle \alpha ={{120}^{\circ }}$

Step 2: Another angle that has the same cosine is $\displaystyle -\alpha =-{{120}^{\circ }}$

Step 3: All the angles that has the cosine $\displaystyle -\frac{1}{2}$ are :

$\displaystyle x=k\cdot {{360}^{\circ }}+{{120}^{\circ }}$ and $\displaystyle x=k\cdot {{360}^{\circ }}-{{120}^{\circ }}$

or $\displaystyle x=2k\pi +\frac{{2\pi }}{3}$ and $\displaystyle x=2k\pi -\frac{{2\pi }}{3}$

Example 4: Solve the equation $\displaystyle \cos (-2x)=\frac{1}{2}$.

$\displaystyle \alpha =6{{0}^{{}^\circ }}$

$\displaystyle x=k\cdot-{{180}^{\circ }}-{{30}^{\circ }}$ o$\displaystyle -2x=k\cdot {{360}^{\circ }}-{{60}^{\circ }}$ and $\displaystyle -2x=k\cdot {{360}^{\circ }}+{{60}^{\circ }}$

$\displaystyle x=k\cdot -{{180}^{\circ }}+{{30}^{\circ }}$

Note! By substituting k with different integers we get the solutions of our equation above:

$\displaystyle k=1$, $\displaystyle \left\{ {{{{480}}^{\circ }}{{{,240}}^{\circ }}} \right\}$

$\displaystyle k=-1$, $\displaystyle \left\{ {-{{{240}}^{\circ }},-{{{480}}^{\circ }}} \right\}$

How to solve the equation $\displaystyle \tan x=c$?

Step 1: We find an angle $\displaystyle \alpha$ that its tangents is equal with $\displaystyle c$.

Step 2: All the angles that has the tangent $\displaystyle c$ are:

$\displaystyle x=k\cdot {{180}^{{}^\circ }}+\alpha$ o$\displaystyle x=k\pi +\alpha$  $\displaystyle k\in Z$

Example 5: Solve the equation $\displaystyle tgx=\sqrt{3}$.

Step 1: An angle that is known to have the tangent $\displaystyle \sqrt{3}$ is $\displaystyle \alpha ={{60}^{\circ }}$

Step 2: All the angles that has the tangent $\displaystyle \sqrt{3}$ are:

$\displaystyle x=k\cdot {{180}^{\circ }}+\alpha$ o$\displaystyle x=k\pi +\alpha$

Example 6: Solve the equation $\displaystyle tg3x=-1$.

$\displaystyle \alpha =13{{5}^{{}^\circ }}$

$\displaystyle 3x=k\cdot {{180}^{\circ }}+13{{5}^{{}^\circ }}$ and $\displaystyle x=k\cdot {{60}^{\circ }}+{{45}^{\circ }}$

o$\displaystyle 3x=k\pi +\frac{{3\pi }}{4}$ and $\displaystyle x=k\frac{\pi }{3}+\frac{\pi }{4}$