Trig Proofs: Websites for Basic Identities

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Discussion Overview

The discussion revolves around finding resources for proofs of basic trigonometric identities, including the additive formulas for sine and cosine, double angle identities, and the Pythagorean identity. The scope includes theoretical aspects of trigonometry and the definitions of trigonometric functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant requests websites that provide proofs for basic trigonometric identities such as sin(x+y) and cos(2x).
  • Another participant suggests using the Euler formula to derive the additive formulas for sine and cosine, indicating that other identities could be proven using this method, but expresses uncertainty about the identity involving sec(x).
  • A third participant clarifies that sec(x) is the reciprocal of cos(x) and states that the identity tg²x + 1 = sec²x can be derived from the Pythagorean identity.
  • A later post discusses the dependence of trigonometric identities on how sine and cosine are defined, mentioning various definitions including those based on right triangles and differential equations.
  • One participant proposes that certain identities may not make sense under specific definitions of sine and cosine, highlighting the importance of definitions in proving these identities.

Areas of Agreement / Disagreement

Participants express differing views on the definitions of sine and cosine and their implications for proving identities. There is no consensus on a single approach or resource for proofs, and multiple perspectives on the definitions and methods remain unresolved.

Contextual Notes

The discussion highlights limitations related to the definitions of trigonometric functions and the assumptions underlying different proof methods. Some identities may depend on specific mathematical frameworks that are not universally accepted.

quasar987
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Does someone know of a website that has proofs for most basic trigonometry identities?

for sin(x+y), cos(x+y), sin(2x), cos(2x), tg²x + 1 = sec²x, etc


thanks!
 
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You can use the Euler formula:
cos(x) + isin(x) = e^(ix)
So:
cos(x+y) + isin(x+y) = e^[i(x+y)]
cos(x+y) + isin(x+y) = e^(ix+iy)
cos(x+y) + isin(x+y) = e^ix * e^iy
Turn the right side into sines and cosines using the original formula and see what you get.

This way you can prove the additive formulas, and the other identities like sin(2x) or cos(x)+cos(y) or tg(x+y) can be proven using them. I don't know about the last one you mentioned since I have no idea what sec(x) is.
 
sec(x) is 1/cos(x). The last identity follows very easily from the Pythagorean identity, sin^2(x) + cos^2(x) = 1.
 
quasar987 said:
Does someone know of a website that has proofs for most basic trigonometry identities?

for sin(x+y), cos(x+y), sin(2x), cos(2x), tg²x + 1 = sec²x, etc


thanks!

That depends strongly on how you define sine and cosine.

If you define them by the "elementary" right triangle ratios, sin2x+ cos2x= 1 follows from the Pythagorean theorem and tan2 x+ 1= sec2 x follows from that. However, such things as sin(x+y), cos(x+y) etc. may not even make sense.

If you define them by sin(x)= (eix- e-ix)/(2i) and cos(x)= (eix+eix)/2, then pig's method can be used.

If you define them by "sin(x) is the function, y, satisfying y"= -y, y(0)= 0, y'(0)= 1" and "cos(x) is the function, y, satisfying y"= -y, y(0)= 1, y'(0)= 0"
Then you can show that any function satisfying y"= -y, y(0)= a, y'(0)= b must be y= a cos(x)+ b sin(x). In particular, for example, cos(x+a) satisfies
y"= -y, y(0)= cos(a), y'(0)= -sin(a) so cos(x+a)= cos(a)cos(x)- sin(a)sin(x) and, with x= b, cos(a+b)= cos(a)cos(b)- sin(a)sin(b).
 

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