Question

Find all solutions in the interval [0, 2π).
sec^2x - 2 = tan^2x

Answer 1

@mathmale

Answer 2

hint: tan^2x = sec^2x - 1

Answer 3

Howdy.
I'd suggest that you use an appropriate identity to re-write this expression so that it's all in the secant function or all in the tangent function.

Note that (tan x)^2 + 1 = (sec x)^2.

Thus, your original equation becomes

(tan x)^2 + 1 - 2 = (tan x)^2,

which reduces to 1=2,

which has NO SOLUTION.

Answer 4

Oh, okay. I thought that one looked funny. But what if it does have a solution? Like this one:\[\sin x =\frac{ \sqrt{3} }{ 2 }\]

Answer 5

@mathmale

Answer 6

I'd suggest that you draw the 2 basic "special triangles," one the 30-60-90 degree triangle, the other the 45-45-90 triangle.

If sin x = Sqrt(3)/2, can you find x from the 30-60-90 degree triangle?
Of course there are other ways to determine x. Which ones ahve you used in the past?

Answer 7

I don't think I've done a question like this before.. So I draw some triangles..

Answer 8

How would knowing the sides help me find the theta angle thing?

Answer 9

In the past, how have you found the sines of given angles?

Let me draw the 30-60-90 triangle for you quickly.

Answer 10

Like, how do you mean.. Like sin(45)? or sin(x)=blahh?

Answer 11

I'll be right back. See whether your can answer your own question:

If sin x = Sqrt(3)/2, what is x?

Answer 12

TIB: Let's take \[\sin x =\frac{ opp }{ hyp }\]\[\sin x = \frac{ opp }{ hyp }=\frac{ \sqrt{3} }{ 2 }.\] and extend it to read

Answer 13

sorry those lines are not in proper order. Nevertheless you can see tht the side opposite to the angle in question is Sqrt(3) and the hypotenuse is 2. Can you identify the angle?

Answer 14

I got that far. But that's why I asked how knowing that would give us the theta angle thing?

Answer 15

Because opposite side Sqrt(3) is the angle we wanted, 60 degrees.
If you're not comfortable with this method, there are certainly others that we could use instead, including using a calculator.

Answer 16

Since I'm used to working with angles and intervals, I'd see right away that 60 deg or pi/3 is one solution. Then I'd look to determine whether there are other angles in the interval [0,2pi) at which the sine is also = to Sqrt(3)/2.