I badly with general answers to trig equations. (Using identities)

[Tyrion101]

I'm completely lost here. I've got the cheat sheet of trig rules, but they don't appear to be helping me, I've watched a half dozen videos on each of cos sin and tan, and nearly all of them discuss the wrong topic. I don't want help on any single problem, but advice. How can I make sense of the trig equations in relation to the trig identities? I realize there is supposed to be a pattern of some sort but I do not see it. I'm close to telling my professor I just give up, and dropping out completely.

 

[Mark44]

I'm completely lost here. I've got the cheat sheet of trig rules, but they don't appear to be helping me, I've watched a half dozen videos on each of cos sin and tan, and nearly all of them discuss the wrong topic. I don't want help on any single problem, but advice. How can I make sense of the trig equations in relation to the trig identities?

I don't know what you mean. Can you give an example where you're having trouble?

I realize there is supposed to be a pattern of some sort but I do not see it. I'm close to telling my professor I just give up, and dropping out completely.

 

[Tyrion101]

Well just about every problem I come to. I'm confused about everything that isn't obviously straight algebra.

 

[Mark44]

Well just about every problem I come to. I'm confused about everything that isn't obviously straight algebra.

How about one for starters? I can't give you any help or advice if I don't know what problems you're having.

 

[Tyrion101]

One that's very confusing is tan^2(2x) = 3 a video I just watched basically ignored the 2x and solved it that way, they replaced 2x with theta, this doesn't really make much sense to me. Can you just ignore that part of the equation like that without putting it back in later?

 

[symbolipoint]

Much of Trigonometry lessons depends on a unit circle, so your primary identity is the equation for a unit circle, from which many other identities can be derived or proved.
[tex]cos^2(x)+sin^2(x)=1[/tex]

Using that, if you understand the equation for a circle, as learned in "Intermediate Algebra", then you will make progress in studying Trigonometry.

 

[Mark44]

One that's very confusing is tan^2(2x) = 3 a video I just watched basically ignored the 2x and solved it that way, they replaced 2x with theta, this doesn't really make much sense to me. Can you just ignore that part of the equation like that without putting it back in later?

No, you can't ignore part of it, but you can replace 2x by ##\theta##, find solutions, and then later replace ##\theta## by 2x.

##tan^2(2x) = 3##
Let ##\theta## = 2x
So ##tan^2(\theta) = 3 \Rightarrow tan(\theta) = \pm \sqrt{3}##
The last equation is actually two equations. Can you solve them? You should be able to solve them exactly (i.e., without a calculator).

 

[symbolipoint]

Much of Trigonometry lessons depends on a unit circle, so your primary identity is the equation for a unit circle, from which many other identities can be derived or proved.
[tex]cos^2(x)+sin^2(x)=1[/tex]

Using that, if you understand the equation for a circle, as learned in "Intermediate Algebra", then you will make progress in studying Trigonometry.

Tyrion,
Can you see how right triangles and circles are related?
tangent means [itex]sin(x)/cos(x)[/itex].

 

[Tyrion101]

That bit I get, I think what I'm lost at is what to do with the 2x.

 

[Mark44]

That bit I get, I think what I'm lost at is what to do with the 2x.

Can you continue from where I stopped in post #7? Don't worry about the 2x yet.

 

[Tyrion101]

If I understand correctly it is pi/3, 2(pi)/3, am I missing part of the answer?

 

[Mark44]

If I understand correctly it is pi/3, 2(pi)/3

##\pi/3## - yes for one of the equations, but there is one other angle in ##[0, 2\pi]##
##2\pi/3## - yes for the other equation, but there is one other angle in ##[0, 2\pi]##.

What are the other two missing angles?

 

[Tyrion101]

4(pi)/3 and 5(pi)/3?

 

[Mark44]

4(pi)/3 and 5(pi)/3?

Yes.

As equations, and including all possible angles, these would be
##\theta = \pi/3 + k(\pi)##
or ##\theta = 2\pi/3 + k(\pi)##
with ##k \in \mathbb{Z}##, the integers.

Now you're ready to undo the substitution.

 

[symbolipoint]

If I understand correctly it is pi/3, 2(pi)/3, am I missing part of the answer?

Are you comfortable either using identities you find (looked up) or identities you are told to try to use? The example you suggest could use a tangent squared identity and a tangent double angle identity.

[itex]1+tan^2(x)=sec^2(x)[/itex] and [itex]tan(2x)=\frac{2tan(x)}{1-tan^2(x)}[/itex]

 

[Tyrion101]

I'm aware of them, I've used the first identity, what do I do to undo the substitution? Multiply all the angles by 2?

 

[symbolipoint]

I'm aware of them, I've used the first identity, what do I do to undo the substitution? Multiply all the angles by 2?

Do you mean, the secant and the tangent? Something other?
[tex]sec(x)=\frac {1}{cos(x)}[/tex]
That is a definition, not identity.

 

[Tyrion101]

Yes.

As equations, and including all possible angles, these would be
##\theta = \pi/3 + k(\pi)##
or ##\theta = 2\pi/3 + k(\pi)##
with ##k \in \mathbb{Z}##, the integers.

Now you're ready to undo the substitution.

I knew how to get to this point, I did not know what to do after.

 

[symbolipoint]

(tan(2x))^2=3
Using tan squared id,
(sec(2x))^2-1=3
(sec(2x))^2=3+1
(sec(2x))^2=4
sec(2x)=+/-2
Using definition
1/cos(2x)=+/-2
cos(2x)=+/-(1/2)

 

[Mark44]

I knew how to get to this point, I did not know what to do after.

Like I saId, now you can undo the substitution.
##\theta = \pi/3 + k(\pi)##
or ##\theta = 2\pi/3 + k(\pi)##

So
##2x = \pi/3 + k(\pi)##
or ##2x = 2\pi/3 + k(\pi)##
It's easy to solve for x.

 

[Tyrion101]

I figured it out before you posted... I feel kind of silly now, guess that happens when you're stuck for a long time on something

 

[Mark44]

(tan(2x))^2=3
Using tan squared id,
(sec(2x))^2-1=3
(sec(2x))^2=3+1
(sec(2x))^2=4
sec(2x)=+/-2
Using definition
1/cos(2x)=+/-2
cos(2x)=+/-(1/2)

This seems a bit roundabout -- to convert to secant, and then to cosine -- when it's not difficult to directly solve the equation using the tangent.

 

[Tyrion101]

Thank you mark you've been a big help to me in my math

 

[Mark44]

I'm aware of them, I've used the first identity, what do I do to undo the substitution? Multiply all the angles by 2?

"Undoing the substitution" means to reverse the step of when you made the substitution.
Early on, we set ##\theta## to 2x. To undo the substitution, just replace ##\theta## by 2x. That's all it is.

 

[Mark44]

Thank you mark you've been a big help to me in my math

You're welcome.

 

[symbolipoint]

This seems a bit roundabout -- to convert to secant, and then to cosine -- when it's not difficult to directly solve the equation using the tangent.

I'm unable to find the more efficient path. I needed the lower level of the identity and the definition, done separately.

 

[Mark44]

I'm unable to find the more efficient path. I needed the lower level of the identity and the definition, done separately.

See post #7.

 

[symbolipoint]

See post #7.

Yes, I could have derived the identity myself. I looked for the identity in wikipedia instead. Deriving it on my own? More low level stuff.

All I keep in my head all the time is the Pythagorean identity and the definitions for the tangent, csc, sec, cot. Also, I usually have Law Of Cosines memorized.

 

[Mark44]

Yes, I could have derived the identity myself. I looked for the identity in wikipedia instead. Deriving it on my own?

I didn't use any identities. It was nothing more than solving the equation tan²(A) = 3, getting ##tan(A) = \pm \sqrt{3}##.

More low level stuff.

All I keep in my head all the time is the Pythagorean identity and the definitions for the tangent, csc, sec, cot. Also, I usually have Law Of Cosines memorized.

 

[symbolipoint]

I didn't use any identities. It was nothing more than solving the equation tan²(A) = 3, getting ##tan(A) = \pm \sqrt{3}##.

Very simple. I looked at the original example, saw two things at once, and then looked for the more complicated route. I would then be dealing the with angle, 2A.

Accidentally looked at post #6 instead of post #7. No need to try to derive anything.

 

[Tyrion101]

I think I understand, so the number of answers varies depending on what function you are using, and I may not understand all of the functions well, however with your info I have figured out that pi/3, and 5pi/3 are the only solutions

 

[symbolipoint]

Just remember, your example was [itex]tan^2(2x)=3[/itex], so after you deal with what is 2x, you want to solve for what is x.

 

[Mark44]

I think I understand, so the number of answers varies depending on what function you are using, and I may not understand all of the functions well, however with your info I have figured out that pi/3, and 5pi/3 are the only solutions

@Tyrion101, no, they are not the only solutions, in fact these aren't even solutions of the original equation, which was ##tan^2(2x) = 3##. Unless there is a restricted domain, which you didn't mention, there are an infinite number of solutions.