Trig Identities to Memorize

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 Joined: Tue Aug 04, 2015 7:59 pm
Trig Identities to Memorize
What trig identities should I have memorized for the GRE subject test? Do I need to know double angle formulas? What about obscure identities involving sec(x), cot(x), etc?
Re: Trig Identities to Memorize
In general, you only need to know three trigonometric identities along with the definitions of tan(x), cot(x), sec(x), csc(x) through sin(x) and cos(x), as well as the fact that cos(x) is an even function: cos(x)=cos(x), and that sin(x) is odd: sin(x)=sin(x). The basic identities are:
(1) The Pythagorean identity: (sin(x))^2+(cos(x))^2=1
(2) The angle sum formula for sine: sin(x+y)=sin(x)cos(y)+cos(x)sin(y)
(3) The angle sum formula for cosine: cos(x+y)=cos(x)cos(y)sin(x)sin(y)
Now, let's play around with these basic identities:
(a) Divide identity (1) by (cos(x))^2 to obtain: (tan(x))^2+1=(sec(x))^2
(b) Divide identity (1) by (sin(x))^2 to obtain: 1+(cot(x))^2=(csc(x))^2
(c) In identity (2), let y=x. We obtain: sin(2x)=2sin(x)cos(x)
(d) In identity (3), let y=x. We obtain: cos(2x)=(cos(x))^2(sin(x))^2
(e) In the RHS of identity (d), replace (sin(x))^2 by 1(cos(x))^2 (from rearranging identity (1)). We obtain: cos(2x)=2(cos(x))^21
(f) Similarly, in identity (d), replace (cos(x))^2 by 1(sin(x))^2 to obtain the alternative formula: cos(2x)=12(sin(x))^2
There are many more identities we can derive by playing around with identities (1)(3), and (a)(f).
(1) The Pythagorean identity: (sin(x))^2+(cos(x))^2=1
(2) The angle sum formula for sine: sin(x+y)=sin(x)cos(y)+cos(x)sin(y)
(3) The angle sum formula for cosine: cos(x+y)=cos(x)cos(y)sin(x)sin(y)
Now, let's play around with these basic identities:
(a) Divide identity (1) by (cos(x))^2 to obtain: (tan(x))^2+1=(sec(x))^2
(b) Divide identity (1) by (sin(x))^2 to obtain: 1+(cot(x))^2=(csc(x))^2
(c) In identity (2), let y=x. We obtain: sin(2x)=2sin(x)cos(x)
(d) In identity (3), let y=x. We obtain: cos(2x)=(cos(x))^2(sin(x))^2
(e) In the RHS of identity (d), replace (sin(x))^2 by 1(cos(x))^2 (from rearranging identity (1)). We obtain: cos(2x)=2(cos(x))^21
(f) Similarly, in identity (d), replace (cos(x))^2 by 1(sin(x))^2 to obtain the alternative formula: cos(2x)=12(sin(x))^2
There are many more identities we can derive by playing around with identities (1)(3), and (a)(f).
Re: Trig Identities to Memorize
(1) and (a)(c) are the only ones that are important in my view. I don't know that the others have ever been necessary on this test.
Re: Trig Identities to Memorize
I've seen an instance of using (2), (3) before on the Subject Math before. Though I have to agree that they are not used often.
Re: Trig Identities to Memorize
I mean it never hurts to memorize more trig stuff. But there's always something more useful to be studying than some trig identity that has very little chance to be used.
I used my approach above. I've TAed Calculus 2 many many times, so none of this stuff was even memorization for me. Some of the others wouldn't be hard for me to rederive if I had to, but certainly I'd rather focus on knowing my stuff on diagonalization than memorizing various formulas.
Edit: If I were going to memorize anything, it would be differential equations stuff. I never studied much of that in college, and memorizing some of those formulas is basically guaranteed to help you solve a problem or two.
I used my approach above. I've TAed Calculus 2 many many times, so none of this stuff was even memorization for me. Some of the others wouldn't be hard for me to rederive if I had to, but certainly I'd rather focus on knowing my stuff on diagonalization than memorizing various formulas.
Edit: If I were going to memorize anything, it would be differential equations stuff. I never studied much of that in college, and memorizing some of those formulas is basically guaranteed to help you solve a problem or two.