View Full Version : How do you verify Trig. Identities?
Doglover2011
Oct 6, 2009, 12:16 PM
I do not understand how to verify Trig. Identities and I have an upcoming test soon. For example, how do I solve this:
Sin^2Theta+Tan^2Theta+Cos^2Theta=Sec^2Theta
This was a problem out of the book, but it's not part of any homework. I didn't really know how to make a problem for this up. I have a list of formulas my teacher gave me, but which ones to use are confusing. Help please?
Perito
Oct 6, 2009, 01:09 PM
Sin^2(\theta)+Tan^2(\theta)+Cos^2(\theta)=Sec^2( \theta)
The usual way to prove these things is to convert everything to sines and cosines and manipulate it until it's obvious (something like "Cos() = Cos()"). Note the following identities that should be memorized because they show up on every test.
Tan(\theta) = \frac {Sin(\theta)}{Cos(\theta)}
Sec(\theta) = \frac {1}{Cos(\theta)}
Cse(\theta) = \frac {1}{Sin(\theta)}
Cot(\theta) = \frac {1}{Tan(\theta)}
Sin^2(\theta) + Cos^2(\theta) = 1
Give it a shot and post your work if you have trouble.
Doglover2011
Oct 6, 2009, 01:29 PM
I'm not sure I understand how to set that up.
Like this?
Cos(pi/2-Theta)+SinTheta/CosTheta+Sin(pi/2-Theta)
Eh, I really don't think I set it up right? I'm using the formulas my teacher gave us, by the way.
Doglover2011
Oct 6, 2009, 01:31 PM
Also, what do I do with the exponents? (by the way, thanks for helping me I really appreciate it!)
Perito
Oct 6, 2009, 07:43 PM
Sin^2(\theta)+Tan^2(\theta)+Cos^2(\theta)=Sec^2( \theta)
rearrange to make it clearer
[Sin^2(\theta)+Cos^2(\theta)] + Tan^2(\theta)=Sec^2( \theta)
using the identities I gave above, you get:
1+\frac {Sin^2(\theta)}{Cos^2(\theta)}=\frac {1}{Cos^2( \theta)}
\frac {Cos^2(\theta) + Sin^2(\theta)}{Cos^2(\theta)} = \frac {1}{Cos^2( \theta)}
and therefore
\frac {1}{Cos^2(\theta)} = \frac {1}{Cos^2( \theta)}
QED
Unknown008
Oct 6, 2009, 11:27 PM
Could you mention all the formulae your teacher gave you? Because for identities, I have quite a small lot, for those which do not require double angles... actually 9. And with them, I can prove your identity, faster than what Perito suggested, which I am not saying is bad. Any method you use is good, provided you know what you're doing.
Doglover2011
Oct 7, 2009, 08:50 AM
Actually, she gave us a bunch of formulas:
Reciprocal identities, Tangent and Cotangenet identities, Pythagorean identites, Cofunction identites, and Negative Angle identities.
Do I need to type these out? That's OK, if so. There are like 14 different formulas to choose from. Which one are thinkging of using?
And I need to use a different formula for each part of the problem, right? So, for this one wouldn't I need to pick 3 of these?
Unknown008
Oct 7, 2009, 08:59 AM
Yes. Ok, here I go!
sin^2\theta + tan^2\theta + cos^2\theta = sec^2\theta
You know that sin^2\theta + cos^2\theta = 1
So, sin^2\theta + tan^2\theta + cos^2\theta = sec^2\theta
sin^2\theta + cos^2\theta + tan^2\theta= sec^2\theta
1 + tan^2\theta= sec^2\theta
Isn't that in your list of formulae? :) Well, it's in mine.
Doglover2011
Oct 7, 2009, 09:05 AM
Yes :) Those are in my list.
So, Can I set it up like this:
Sin^2Theta+Cos^2Theta=1+1+Tan^2Theta=Sec^2Theta?
I don't think I did that right though, it looks weird XD
Unknown008
Oct 7, 2009, 09:07 AM
No, you cannot put all of them on a line. The way you put it means:
sin^2\theta+cos^2\theta = 1+1+tan^2\theta
which is false!
You can however write:
sin^2\theta+cos^2\theta +tan^2\theta= 1+tan^2\theta=sec^2\theta
Doglover2011
Oct 7, 2009, 09:48 AM
So, I don't need to use a separate formula for each part of the identity? The sin, cos, and tan?
Is this how your saying you set it up?
Sin^2Theta+Tan^2Theta+Cos^2Theta=Sec^2Theta
1+Tan^2Theta=Sec^2Theta
Then, simpifly from there?
Unknown008
Oct 7, 2009, 09:51 AM
Well, that's how I would have done it... since tan^2theta + 1 = sec^2theta
Doglover2011
Oct 7, 2009, 09:56 AM
Oh, OK I thought so :) Thanks so much for the help!
I was looking at the other formulas and I don't think it can be simpifed anymore, can it? Because I don't see anymore that have tan and sec.
So, that would make tan^2theta + 1 = sec^2theta the answer, correct?
Unknown008
Oct 7, 2009, 10:11 AM
Yes :)
Doglover2011
Oct 7, 2009, 11:26 AM
Ok. Thank you VERY much for taking time out of your day to help me!
Unknown008
Oct 7, 2009, 11:33 AM
You're welcome! :)