Solving Trigonometric Equations She Loves Math

With Graphing:

(Easier!) Note that we use  0 for Xmin

and \(\displaystyle \frac{{2\pi }}{3}\) for

Xmax since \(\displaystyle \frac{{2\pi }}

{3}\) is the period:

Get both intersections and add factors

of \(\displaystyle \frac{{2\pi }}{3}\) (the

period):

\(\displaystyle \left\{ \begin{align}\theta

|\,\theta &=.119+\frac{{2\pi k}}

{3}\\\,\,\,\,\,\,\,\,\,\,\,\,&=\,.548+\frac{{2\pi

k}}{3}\end{align} \right\}\)

Reals}\)

 

Without Graphing:

Work backwards to

get the value of \(x\),

using arctan of \

(\displaystyle \frac{1}

{{10}}\). For the

answer 2.213, add \

(\displaystyle

\frac{{5\pi }}{2}k\)

(period) to get all the

solutions.

With Graphing:

Note that we use  0 for

Xmin and \

(\displaystyle

\frac{{\,\pi }}{{\left(

{\frac{2}{5}}

\right)}}=\frac{{5\pi }}

{2}\) for Xmax since \

(\displaystyle

\frac{{5\pi }}{2}\) is the

period:

 

Add factors of \

(\displaystyle

\frac{{5\pi }}{2}\) (the

period):

\(\displaystyle \left\{

{x|\,x=2.213+\frac{{5\pi

k}}{2}} \right\}\)