Trigonometric equations
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.
Solving Trigonometric Equations
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, the table of the angles 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 }}$ or $ \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 $ or $ \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 $ or $ \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 }}$
or $ \displaystyle 3x=k\pi +\frac{{3\pi }}{4}$ and $ \displaystyle x=k\frac{\pi }{3}+\frac{\pi }{4}$
