Question 1088061
.
Find all solutions in the interval [0,360)
cos^2(theta)+8cos(theta)+5=0
~~~~~~~~~~~~~~~~


<pre>
{{{cos^2(theta) + 8cos(theta) + 5}}} = 0.

Introduce new variable x = {{{cos(theta)}}}. Then your equation takes the form

{{{x^2 + 8x + 5}}} = 0.


Apply the quadratic formula to solve for x:

{{{x[1,2]}}} = {{{(-8 +- sqrt(8^2 - 4*1*5))/2}}} = {{{(-8 +- sqrt(44))/2}}} = {{{-4 +- sqrt(11)}}}.


1)  {{{x[1]}}} = {{{-4 + sqrt(11))}}} =~ -0.68.

    {{{cos(theta)}}} = {{{-4 + sqrt(11)}}}  ====>  {{{theta}}} = {{{arccos(-4 + sqrt(11))}}}  and  {{{theta}}} = {{{2pi - arccos(-4 + sqrt(11))}}}.



2)  {{{x[2]}}} = {{{-4 - sqrt(11))}}} < -1.

    There is NO {{{theta}}} satisfying {{{cos(theta)}}} = {{{x[2]}}}.


<U>Answer</U>. The original equation has two solutions  {{{theta}}} = {{{arccos(-4 + sqrt(11))}}}  and  {{{theta}}} = {{{2pi - arccos(-4 + sqrt(11))}}}.
</pre>

Solved.



If you want to see more solved problems/samples on trigonometric equations, look into the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solving-simple-problems-on-trigonometric-equations.lesson>Solving simple problems on trigonometric equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solving-typical-problems-on-trigonometric-equations.lesson>Solving typical problems on trigonometric equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solving-more-complicated-problems-on-trigonometric-equations.lesson>Solving more complicated problems on trigonometric equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solved-problems-on-trigonometric-equations.lesson>Solving advanced problems on trigonometric equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/OVERVIEW-of-lessons-on-calculating-trig-functions-and-solving-trig-equations.lesson>OVERVIEW of lessons on calculating trig functions and solving trig equations</A>

in this site.



Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic "<U>Trigonometry: Solved problems</U>".