Question 239784
Before we start a few of notes:<ul><li>Algebra.com's formula software for some reason does not "do" theta. So I will be using just "t" instead of theta.</li><li>Algebra.com's formula software does not handle powers of functions well. You may see some multiplication symbols, "*", between the function and the arguments. These multiplication symbols are not correct and should not be there.</li><li>There are two kinds of equations in Trig.:<ul><li>Identities which are equations which are true for all possible angles. These are the equations you "prove".</li><li>Conditional equations which are true for only some angles. These equations you solve to find the angles which make the equation true.</li></ul>The equation you have in this problem is a conditional equation not an identity. So we will not prove it. We will solve it.</li></ul>
Whether you are proving an identity or trying to solve a conditional equation, if the path is not clear to you it is often a good idea to change other Trig functions into sin's and cos's. This is where we will start. Since cot(t) = cos(t)/sin(t)
{{{cos^2(t)/ (cot^2(t) - cos^2(t)) = 3}}}
becomes, after replacing {{{cot^2(t)}}}:
{{{cos^2 (t)/ ((cos^2 (t)/sin^2(t)) - cos^2 (t)) = 3}}}
Since every term in both the numerator and denominator of the fraction has {{{cos^2(t)}}} in it, the fraction will simplify greatly if we multiply the numerator and denominator by its reciprocal:
{{{(cos^2 (t)/ (cos^2 (t)/sin^2(t) - cos^2 (t)))((1/cos^2(t))/(1/cos^2(t))) = 3}}}
All the {{{cos^2(t)}}}'s cancel leaving:
{{{1/(1/sin^2(t) - 1) = 3}}}
Now we can further simplify the left side by multiplying the numerator and denominator by {{{sin^2(t)}}}:
{{{(1/(1/sin^2(t)) - 1)(sin^2(t)/sin^2(t)) = 3}}}
which gives us:
{{{sin^2(t)/(1-sin^2(t)) = 3}}}
The denominator should look familiar. It is {{{cos^2(t)}}}:
{{{sin^2(t)/cos^2(t) = 3}}}
The left side is {{{tan^2(t)}}}:
{{{tan^2(t) = 3}}}
We now have the equation in a form we can solve. (Up to this point, all we've been doing is "playing around" with the equation, using Algebra and Trig. properties/identities, until we find an equation we can solve.)<br>
If {{{tan^2(t) = 3}}} then
{{{tan(t) = sqrt(3)}}} or {{{tan(t) = -sqrt(3)}}}
If you know your special angle values well you will recognize that the reference angle is 60 degrees or {{{pi/3}}} radians. Since theta is normally used for angles measured in degrees, we will use 60 degrees. So the solution is:
{{{t = 60 + 180n}}} or {{{t = -60 + 180n}}}
Notes about the solution:<ul><li>The "+ 180n"<ul><li>Trig functions are periodic and so when finding angles, there will be an infinite number of solutions. And our solution should reflect <i>all</i> the angles which fit. The "+ 180n" is used as a way to specify this infinite set of angles.</li><li>The "n" can be any integer.</li><li>The "180" is used because the period of the tan function is 180 degrees. (For sin, cos, sec and csc we would use 360.)</li></ul></li><li>The "60". Any angle with a tan of {{{sqrt(3)}}} could be used. 60 is the most "obvious".</li><li>The "-60". Any angle with a tan of {{{-sqrt(3)}}} could be used. -60, 120 or 300 are probably the most "obvious".</li></ul>