Question 677360
tan(x)sec(x)-2tan(x)=0
Solving these equations usually starts with using algebra and/or Trig. identities to transform the equation so that you have one or more equations of the form:
trigfunction(expression) = number<br>
With this equation the transformation is not too difficult. First we factor out tan(x):
tan(x)(sec(x)-2)=0
And then use the Zero Product Property:
tan(x) = 0 or sec(x)-2 = 0
Adding 2 to each side of the second equation we get:
tan(x) = 0 or sec(x) = 2
We now have two equations of the desired form.<br>
The next step is to write the general solution for each equation. The general solution expresses <i>all</i> the solutions to the equations. We'll start with:
tan(x) = 0
We should recognize that 0 is a special angle value for tan. So we will not need our calculators. tan is zero angles of 0, {{{pi}}} and for all other angles that are co-terminal with these two. We express this with:
{{{x = 0 + 2pi*n}}}
{{{x = pi + 2pi*n}}}
The n's in these equations represent any integer. By replacing "n" with an integer you end up with an x that is 0, {{{pi}}} or some other angle that is co-terminal.<br>
Next
sec(x) = 2
You may not recognize that 2 is a special angle value for sec. Since cos is the reciprocal of sec, an angle whose sec is 2 will have a cos of 1/2:
{{{cos(x) = 1/2}}}
We should recognize that 1/2 is definitely a special angle value for cos. The reference angle is {{{pi/3}}}. And since the 1/2 is positive and cos (and sec) are positive in the 1st and 4th quadrants, the angles we are looking for will terminate in either the 1st or 4th quadrant and have a reference angle of {{{pi/3}}}:
{{{x = pi/3 + 2pi*n}}}
{{{x = -pi/3 + 2pi*n}}} (or {{{x = 5pi/3 + 2pi*n}}})<br>
Altogether the general solution to your equation is:
{{{x = 0 + 2pi*n}}}
{{{x = pi + 2pi*n}}}
{{{x = pi/3 + 2pi*n}}}
{{{x = -pi/3 + 2pi*n}}} (or {{{x = 5pi/3 + 2pi*n}}})<br>
Many, but not all, of these problems ask you to find a specific solution. For example: "Find the least positive solution to ..." or "Find all solutions to ... that are between zero and {{{2pi}}}". When specific solutions are requested you use the general solution equation(s) and various integer values for "n" until you have found the requested solution(s).<br>
In this case the problem does not ask for a specific solution. It asks for "all real values". So the general solution is the solution when no specific solution is requested.<br>
P.S. The period for tan (and cot) is just {{{pi}}}. So instead of 2 general solution equations for tan (or cot) equations, you can just use one. In this case
{{{x = 0 + 2pi*n}}}
{{{x = pi + 2pi*n}}}
can be replaced with just:
{{{x = 0 + pi*n}}}
(Note how the {{{2pi}}} has been replaced with just {{{pi}}}.)