Math Is Fun Forum

denis_gylaev

Member

Registered: 2015-03-19

Posts: 66

Trigonometry Proof

Explain why we must have

for any

such that

is not an integer multiple of

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Trigonometry Proof

hi denis_gylaev

Start with Pythagoras: opp^2 + adj^2 = hyp^2 and divide by opp^2.

This shows it works for angles in any right angled triangle ie 0 - 90 degrees.  The more general result follows because when an angle is between 90 and 180, or over 180, there is always an acute angle such that sine^2(bigger angle) = sine^2(acute angle) and similarly for cosine and tangent.  So the acute angled formula also works for other angles.  The reason there is always such an acute angle follows from the more general definition of sine / cosine  /  tangent.

Multiples of 180 are excluded to avoid division by zero.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob