# How do you find sinx=1/2?

Apr 14, 2015

Use the trig conversion table and the trig unit circle to solve $\sin x = \frac{1}{2.}$
Trig table gives $\sin x = \frac{1}{2} = \sin \left(\frac{\pi}{6}\right) - \to {x}_{1} = \frac{\pi}{6}$.
Trig circle gives another arc ${x}_{2} = 5 \frac{\pi}{6}$ that has the same sin value $\left(\frac{1}{2}\right)$.
Since $f \left(x\right) = \sin x$ is a periodic function, with period $2 \pi$, then there are an infinity of arcs that have the same sin value $\left(\frac{1}{2}\right)$, when the variable arc x rotates around the trig unit circle many times. They are called "extended answers".

They are: $x = \frac{\pi}{6} + K \cdot 2 \pi$; and $x = 5 \frac{\pi}{6} + K \cdot 2 \pi .$ ($K$ is a whole number)

Apr 19, 2015

*Diagram not drawn to scale.

Mar 15, 2017

${30}^{\circ}$ $\mathmr{and} {150}^{\circ}$

#### Explanation:

We can find the answer using a triangle or the unit circle

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Using a right triangle

color(blue)(sin(theta)=1/2

As we see in the diagram, $\sin \left({30}^{\circ}\right)$ has a opposit and hypotenuse $1 \mathmr{and} 2$

So,

color(green)(rArrsin(30^circ)=1/2

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Using unit circle

As ${30}^{\circ}$ and ${150}^{\circ}$ has a $\sin$ of $\frac{1}{2}$,

color(green)(rArrsin(30^circ)=1/2

color(green)(rArrsin(150^circ)=1/2

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Hope this helps! :)