solving trig identities. is there a trick like steps to follow?

Question

my teacher told us it was random... but there should be a standard... like from a complicated to simplified... or a specific pattern to look out for...
is there stuff like that?
plz don't make it too complicated i m only in precal... so we only deal with the basic trig functions
an example question is like



(1+sinx + cosx)/(1+sinx-cosx) = (1+cosx)/sinx

its a problem the professor went through in class.. just want to give u a sample of the problem level
its college level not high school so yeh its alil tougher >.<

oh yeh no cross multiplying... eh... x/2? >.> iono about that...

Answer 1

Often it's a case of rearranging into simpler forms, as you would with any equation. You should also look for patterns and if there's anyway to use an existing trig identity (to simplify).

Eliminate any quotients first.

If you have something in terms of tan's, sec's, etc - then it's _sometimes_ easier if you write these all in terms of just sin's and cos's.


Generally though, it's just a case of practise - lots and lots. And you will eventually get the hang of it, and be able to predict what steps to take well in advance.

Answer 2

To show the one you have, I would first try to 'cross multiply' and expand both sides. Then, I would expect the identity sin^2 x +cos^ x=1 to come up, so I would look for sin^2 x and cos^2 x on either side and see what I could work with.

In your case, it looks like

(1+sinx +cosx)/(1+sinx-cosx)=?
(1+cosx)/sinx

(1+sinx+cosx)*(sinx)=?
(1+cosx)*(1+sinx-cosx)

sinx+sin^2 x+sinx cosx=?
1+sinx-cosx+cosx+sinx cosx -cos^2 x

Now simplify the right side by cancelling the cosx
sinx+sin^2 x+sinx cosx=?
1+sinx+sinx cosx-cos^2 x

But now notice that the sinx and the sinx cosx term are the same on both sides, so we are really asking if
sin^2 x=1-cos^2 x, which we know is true by a pythagorean identity.

The steps to take do depend on the specific problem, but you shold be very aware of all the basic identities (pythogorean, double angles, etc). A trick that is often a good one is to wite everything as sine's and cosine's. So if you have tan x and don't see anything to do, write it as (sin x)/(cos x) and work from there. Algebra skills are essential also.

Answer 3

Trigonometric identities are somewhat difficult yet I doubt there are a number of tricks that may assist you out with them, or a minimum of ive in no way used any. the final factor to do for my area is make flash playing cards with diverse identities on them or perhaps the addition, subtraction, multiplication, and branch identities so which you will keep going over them and getting embedded on your head. on the least basically keep in mind the consumer-friendly equation sin^2(x) + cos^2(x) = a million you are able to derive like 12 of the different identities from that one.

Answer 4

Get rid of denominators by multiplying

(1 + sinx + cosx)(sinx) = (1 + cosx)(1 + sinx - cosx)

sinx + sin^2x + sinxcosx = 1 + sinx - cosx + cosx + sinxcosx - cos^2x

sinx + sin^2x = 1 + sinx - cosx + cosx - cos^2x

sinx + sin^2x = 1 + sinx - cos^2x

sin^2x = 1 - cos^2x

This is just another way of stating the Pythagorean theorem.

Answer 5

I cannot give you any general steps to follow, unfortunately. However, I would stress the importance of knowing the following formulae, which are a slight re-arrangement of the expressions for cos(2x) in terms of cos(x) or sin(x):
1 + cos(x) = 2cos^2(x/2)
1 - cos(x) = 2sin^2(x/2).
The expressions 1 +/- cos(x) occur very frequently, and if you learn only
cos(2x) = cos^2(x) - sin^2(x)
cos(2x) = 2cos^2(x) - 1
cos(2x) = 1 - 2sin^2(x)
you will probably not know what to do with 1 +/- cos(x) when you come across it.

Now, do they work on this identity?

LHS = [ 2cos^2(x/2) + 2sin(x/2)cos(x/2) ]
/ [ 2sin^2(x/2) + 2sin(x/2)cos(x/2) ]
= 2cos(x/2) [ cos(x/2) + sin(x/2) ]
/ 2sin(x/2)[ sin(x/2) + cos(x/2) ]
= cot(x/2).

RHS = 2cos^2(x/2) / 2sin(x/2)cos(x/2)
= cot(x/2).

They do.

Answer 6

I agree with Ozo. Its like trying to teach someone to ride a bicycle or water ski. Whatever they tell you is going to sound great, but nothing substitutes for practice. Like the examples above, one day, You'll just 'get it'.