Solving Trigonometric Equations

You can use both Exact and Numeric from the Solve submenu to find solutions to trigonometric equations. These operations also convert degrees to radians. Use of decimal notation in the equation gives you a numerical solution, even with Exact.

Equation Solve + Exact Solve + Numeric
x = sin${\frac{\pi}{4}}$ x = ${\frac{1}{2}}$$\sqrt{{2}}$ x = .70711
sin 22o = ${\dfrac{{14}}{c}}$ $\;\stackrel{{\vspace{1pt}}}{{c=\dfrac{14}{\sin
\frac{11}{90}\pi }}}\;$ c = 37.373
log10sin x = - 1.1679 x = 6.7988×10-2 x = 3.0736
x = 3o54 x =  ${\frac{{13}}{{600}}}$π x =  6.8068×10-2 6pt

Note that the answers are different for the equation log10sin x = - 1.1679. This difference occurs because there are multiple solutions and the two commands are finding different solutions. The Numeric command from the Solve submenu offers the advantage that you can specify the range in which you wish the solution to lie. Enter the equation and the range in different rows of a display or a one-column matrix.

$\blacktriangleright$ Solve + Numeric

x = 10 sin x
x$\left(\vphantom{ 5,\infty }\right.$5,∞$\left.\vphantom{ 5,\infty }\right)$
, Solution is: $\left\{\vphantom{ x=7.0682}\right.$x = 7.0682$\left.\vphantom{ x=7.0682}\right\}$

The interval $\left(\vphantom{ 5,\infty }\right.$5,∞$\left.\vphantom{ 5,\infty }\right)$ was specified for the solution in the preceding example. By specifying other intervals, you can find all seven solutions: $\left\{\vphantom{ x=0}\right.$x = 0$\left.\vphantom{ x=0}\right\}$, $\left\{\vphantom{ x=\pm 2.8523}\right.$x = ±2.8523$\left.\vphantom{ x=\pm 2.8523}\right\}$, $\left\{\vphantom{
x=\pm 7.0682}\right.$x = ±7.0682$\left.\vphantom{
x=\pm 7.0682}\right\}$, $\left\{\vphantom{ x=\pm 8.4232}\right.$x = ±8.4232$\left.\vphantom{ x=\pm 8.4232}\right\}$, as depicted in the following graph. The Exact command for solving equations gives only the solution x = 0 for this equation.dtbpF3in2.0003in0ptFigure

Scientific Notebook automatically converts degrees to radians when any operation is applied. To go in the other direction, replace 2π radians with 360o and convert other angles proportionately. You can also solve directly for the number of degrees. Both methods follow.

$\blacktriangleright$ To convert radians to degrees

1.
Write the equation ${\dfrac{{\theta }}{{360}}}$ = ${\dfrac{{x}}{{2\pi }}}$, where x represents radians.

2.
Leave the insertion point in this equation.

3.
From the Solve submenu, choose Exact or Numeric.

4.
Name θ as the Variable to Solve for.

5.
Choose OK to get θ = ${\dfrac{{180}}{{\pi }}}$x.

$\blacktriangleright$ To convert radians to degrees

  1. Write an equation such as 2 = θo

  2. Leave the insertion point in this equation.

  3. From the Solve submenu, choose Exact or Numeric.


\begin{example}
Convert $x=\dfrac{13}{600}\pi \,$\ radians to degrees as follow...
...^{\prime }=\,\frac{13}{600}\pi
\end{displaymath}
\end{itemize}
\end{example}