Q 134 - Precalculus Enhanced with Graphing Utilities

Question

Explain how to find the value of $\sin {390}^o$ using periodic properties. 

Step-by-Step Solution

Answer

The value of $\sin {390}^o$ is :-

$\frac{1}{2}$

Step 1. Given Information

We have to find the value of $\sin {390}^o$.

We will use periodic properties of sine function to find the value of $\sin {390}^o$.

Step 2. To find the value

We can write ${390}^o$ as a sum of ${360}^o$ and ${30}^o$.

This gives us :-

$\sin \left({390}^o\right)=\sin \left({360}^o+{30}^o\right)$.

As we know $\pi ={180}^o$, that is ${360}^o=2\pi$.
Then we have :-

$\sin \left({390}^o\right)=\sin \left(2\pi +{30}^o\right)$.

We also know that sine function is a periodic function of period $2\pi$.

That is :-

$\sin (2\pi +\theta )=\sin \theta$.

By using this periodic property we have :-

$\sin \left({390}^o\right)=\sin {30}^o$

We know that the value of $\sin {30}^o=\frac{1}{2}$.

By putting this value, we have :-

$\sin {390}^o=\frac{1}{2}$

This is the required value.