# Thread: Trig equation and identities

1. ## Trig equation and identities

Sitting here studying for a final exam this week, and I am SOO bad at identities. Maybe I just don't know them well enough.
Anyhow,

Find the solutions to the following:
$\displaystyle sec^2x = 2secx$

I got this far...or maybe it's all wrong, I duno.
$\displaystyle 1+tan^2x = 2secx$ * from a Pythagorean identity
$\displaystyle 1+tan^2x = \frac{1}{2cosx}$ * from the reciprocal of sec?
$\displaystyle 1+tan^2x (cos2x-sin2x)$ *cross multiplied and double angle formula? I feel this wasn't a good move, haha.

2. Originally Posted by tiar
Sitting here studying for a final exam this week, and I am SOO bad at identities. Maybe I just don't know them well enough.
Anyhow,

Find the solutions to the following:
$\displaystyle sec^2x = 2secx$

I got this far...or maybe it's all wrong, I duno.
$\displaystyle 1+tan^2x = 2secx$ * from a Pythagorean identity
$\displaystyle 1+tan^2x = \frac{1}{2cosx}$ * from the reciprocal of sec?
$\displaystyle 1+tan^2x (cos2x-sin2x)$ *cross multiplied and double angle formula? I feel this wasn't a good move, haha.

I would solve it as a quadratic rather than using an identity:

sec^2(x) - 2sec(x) = 0

sec(x)(sec(x)-2) = 0

sec(x) = 0 or sec(x) = 2

Can you finish from there?

3. Originally Posted by e^(i*pi)
I would solve it as a quadratic rather than using an identity:

sec^2(x) - 2sec(x) = 0

sec(x)(sec(x)-2) = 0

sec(x) = 0 or sec(x) = 2

Can you finish from there?
Oh wow, that's much simpler.
Further questions...
So sec(x) = 0....1/cosx = 0...when cosine is zero, it's between pi/2 and 3pi/2, right?
So is x=1? I think I'm a bit confused here.

sec(x)=2
1/cosx = 2
No solution for this, right? Outside of unit circle?

Hopefully I'm not too far off...:\

4. Originally Posted by tiar
Oh wow, that's much simpler.
Further questions...
So sec(x) = 0....1/cosx = 0...when cosine is zero, it's between pi/2 and 3pi/2, right?
So is x=1? I think I'm a bit confused here.

sec(x)=2
1/cosx = 2
No solution for this, right? Outside of unit circle?

Hopefully I'm not too far off...:\
That sounds right, there is no determined solution for sec(x) = 0

For sec(x) = 2 there is a solution: you can take the reciprocal of both sides:

$\displaystyle \frac{1}{cos(x)} = 2$

$\displaystyle \frac{1}{\frac{1}{cos(x)}} = \frac{1}{2}$

$\displaystyle cos(x) = 0.5$

$\displaystyle x = \frac{\pi}{3} \text { and } \frac{4\pi}{3}$

those are the two solutions between 0 and 2pi but in general it will be

$\displaystyle x = \frac{\pi}{3} \pm k\pi$ where k is an integer