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

1. Feb 24, 2015

### 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.

2. Feb 24, 2015

### Staff: Mentor

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

3. Feb 24, 2015

### Tyrion101

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

4. Feb 24, 2015

### Staff: Mentor

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

5. Feb 24, 2015

### 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?

6. Feb 24, 2015

### 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.
$$cos^2(x)+sin^2(x)=1$$

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

7. Feb 24, 2015

### Staff: Mentor

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).

8. Feb 24, 2015

### symbolipoint

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

9. Feb 24, 2015

### Tyrion101

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

10. Feb 24, 2015

### Staff: Mentor

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

11. Feb 24, 2015

### Tyrion101

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

12. Feb 24, 2015

### Staff: Mentor

$\pi/3$ - 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?

13. Feb 24, 2015

### Tyrion101

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

14. Feb 24, 2015

### Staff: Mentor

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.

15. Feb 24, 2015

### symbolipoint

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.

$1+tan^2(x)=sec^2(x)$ and $tan(2x)=\frac{2tan(x)}{1-tan^2(x)}$

16. Feb 24, 2015

### 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?

17. Feb 24, 2015

### symbolipoint

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

18. Feb 24, 2015

### Tyrion101

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

19. Feb 24, 2015

### 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)

20. Feb 24, 2015

### Staff: Mentor

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.