Proving sin^2x + cos^2x = 1 - Which one of my proofs is correct?

Apologies for the bad handwriting. I managed to muster up with method 1 and then saw method 2 which made me wonder which one is the correct proof? In method 1 I'm basically expressing the trig functions in terms of the sides and in method 2 I'm expressing the sides of the triangle using the trig functions. Which one is correct?

@RonnieRJ taggin u cus u stem buddy. help a stem buddy out.

Reply 1

username1560589

20

Original post by High Stakes

Apologies for the bad handwriting. I managed to muster up with method 1 and then saw method 2 which made me wonder which one is the correct proof? In method 1 I'm basically expressing the trig functions in terms of the sides and in method 2 I'm expressing the sides of the triangle using the trig functions. Which one is correct?

@RonnieRJ taggin u cus u stem buddy. help a stem buddy out.

They both are.

Reply 2

TeeEm

19

Original post by High Stakes

Apologies for the bad handwriting. I managed to muster up with method 1 and then saw method 2 which made me wonder which one is the correct proof? In method 1 I'm basically expressing the trig functions in terms of the sides and in method 2 I'm expressing the sides of the triangle using the trig functions. Which one is correct?

@RonnieRJ taggin u cus u stem buddy. help a stem buddy out.

for an elementary proof for 0<x<90 they are both correct.

for any x these are not proofs

Reply 3

High Stakes

OP

18

Original post by TeeEm

for an elementary proof for 0<x<90 they are both correct.

for any x these are not proofs

I see. So for any x, I would have to use complex analysis?

Reply 4

TeeEm

19

Original post by High Stakes

I see. So for any x, I would have to use complex analysis?

usual is to let f(x) = sin²x + cos²x

and differentiate

Reply 5

Healthy_LS

2

Original post by High Stakes

I see. So for any x, I would have to use complex analysis?

Complex analysis? No need for that. You just use symetry, periodocity, and Shifts of trig functions to prove it for all other x. After establising 0<x<90 ofcourse like you did.

Reply 6

rxns_00

15

Original post by High Stakes

Apologies for the bad handwriting. I managed to muster up with method 1 and then saw method 2 which made me wonder which one is the correct proof? In method 1 I'm basically expressing the trig functions in terms of the sides and in method 2 I'm expressing the sides of the triangle using the trig functions. Which one is correct?

@RonnieRJ taggin u cus u stem buddy. help a stem buddy out.

They're both correct but the problem is the identity isn't going to be supported by these methods if the angle isnt acute?

Reply 7

davros

16

Original post by High Stakes

I see. So for any x, I would have to use complex analysis?

Depends what you mean by "for any x" :smile:

If x can be a complex value, then you need a an analytical definition of sin and cos in the first place, so your proof needs to be 'analytical' :smile:

For acute real x, your proof is fine.

For non-acute real x you need to decide how you're going to extend the definition of sin and cos and work from there.

It would be messy, but you could avoid calculus by breaking the problem down into cases e.g. suppose 90 < x < 180. Then x = 90 + y where y is acute. Now use the trig addition formulas to rewrite things in terms of sin y and cos y, and you'll end up using the 'version' of the formula for 2 acute angles.

Reply 8

uncapitalised

9

See this TSR thread for a large number of proofs.

(edited 8 years ago)

Reply 9

Dalek1099

21

Original post by TeeEm

for an elementary proof for 0<x<90 they are both correct.

for any x these are not proofs

These proofs are correct for all angles.OP's x and y are correct and work for all angles due to the sine and cosine definition of the Unit Circle, although x,y and theta have been defined differently by OP to the normal way.You can easily see that if you rotate theta the way OP has defined x and y still holds for non-acute theta.

Reply 10

TeeEm

19

Original post by Dalek1099

These proofs are correct for all angles.OP's x and y are correct and work for all angles due to the sine and cosine definition of the Unit Circle, although x,y and theta have been defined differently by OP to the normal way.You can easily see that if you rotate theta the way OP has defined x and y still holds for non-acute theta.

sure.

Proving sin^2x + cos^2x = 1 - Which one of my proofs is correct?

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