1. ## tan cot sec

solve this equation for A in the range 0<A<360
tanA + cotA = 2secA
i found A= 30, 150, 210, 330 degrees and all of these results fit the tangent function i solved for. but the answer includes only A = 30 and 150 degrees. why is this?

2. Originally Posted by furor celtica
solve this equation for A in the range 0<A<360
tanA + cotA = 2secA
i found A= 30, 150, 210, 330 degrees and all of these results fit the tangent function i solved for. but the answer includes only A = 30 and 150 degrees. why is this?
$\tan{A} + \cot{A} = 2\sec{A}$

$\frac{\sin{A}}{\cos{A}}+ \frac{\cos{A}}{\sin{A}} = \frac{2}{\cos{A}}$

$\frac{\sin^2{A}}{\sin{A}\cos{A}} + \frac{\cos^2{A}}{\sin{A}\cos{A}} = \frac{2\sin{A}}{\sin{A}\cos{A}}$

$\frac{\sin^2{A} + \cos^2{A}}{\sin{A}\cos{A}} = \frac{2\sin{A}}{\sin{A}\cos{A}}$

$\sin^2{A} + \cos^2{A} = 2\sin{A}$

$1 = 2\sin{A}$

$\sin{A} = \frac{1}{2}$

Since $\sin$ is only positive in the first and second quadrants, in the domain $0^{\circ} \leq x \leq 360^{\circ}$, there will only be two solutions.

$A = 30^{\circ}$ or $A = 150^{\circ}$.

3. Hello furor celtica
Originally Posted by furor celtica
solve this equation for A in the range 0<A<360
tanA + cotA = 2secA
i found A= 30, 150, 210, 330 degrees and all of these results fit the tangent function i solved for. but the answer includes only A = 30 and 150 degrees. why is this?
ProveIt has given you the correct solution. But to try to answer your question, did you, in your solution, square both sides of the equation at some point? If you did, that will be why you got the extra, unwanted solutions.

You can see a similar thing if you square both sides in a simple non-trig equation. For example:

$x+1=4$

The solution is, of course, $x = 3$. But if you square both sides:

$(x+1)^2 = 16$

$\Rightarrow x^2 + 2x + 1 = 16$

$\Rightarrow x^2 +2x - 15=0$

$\Rightarrow (x +5)(x-3)=0$

$\Rightarrow x = -5$ or $3$

And the value $x = -5$ doesn't satisfy $x+1 = 4$, but $x+1 = -4$. But of couse, both $4^2$ and $(-4)^2$ give the value $16$.

So, always beware of squaring both sides of an equation: you'll sometimes introduce extra, unwanted solutions if you do.