SOLUTION: Please solve equation for x over the interval [0°,360°). {{3sectheta-2√3=0}} show all work.

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Question 683115: Please solve equation for x over the interval [0°,360°). {{3sectheta-2√3=0}}
show all work.
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!

First, isolate the function. Adding :

And divide by 3:

Perhaps if you know your special angle values very well you might recognize that the right side is a special angle value for sec. If not, then we can convert this to cos. Since cos and sec are reciprocals of each other then if

then

Rationalizing the right side:



This is obviously a special angle value for cos. The references angle whose cos is is 30. Since cos (and sec) are positive in the 1st and 4th quadrants we get a general solution of:


This general solution expresses the infinite number of values for that are solutions to your original equation.

Your problem asks for solutions over the interval [0, 360). For these solutions we take the general solution equations and replace the n's with integers until we are satisfied we have found all the 's in the interval.
From the equation:

if n = 0 then
if n = 1 (or higher integers), is more than 360.
if n = -1 (or any other negative integer), is below 0.
From the equation:

if n = 0 (or any negative integer), is below 0.
if n = 1, is 330.
if n = 2 (or higher integers), is more than 360.
So the only solutions over the interval [0, 360) are 30 and 330.
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