Question

Can someone explain why tan(theta)=sin(theta)/cos(theta)?

Answer 1

sine = opp/hyp
cosine = adj/hyp
tangent = opp/adj

Answer 2

if we divide sine by cosine, we get...

sine/cosine

(opp/hyp) divided by (adj/hyp)

(opp/hyp) * (hyp/adj)

(opp*hyp)/(hyp*adj)

opp/adj

which is equal to tangent

Answer 3

In a right angles triangle:
The Sine of an angle is equal to the ratio of the opposite side to the hypotenuse.
The Cosine of an angle is equal to the ratio the adjacent side to the hypotenuse.

The Tangent is equal to the ratio of the opposite side to the adjacent side.

The ratio of the sine to the cosine is (opp/hypo) / (adj /hypo) = opp/adj

Answer 4

Not that the other answers are'n just fine, but I'd sum it up as saying it's simple a definition.

Answer 5

funny thing though, ever notice that the tangent line to some theta on the unit circle has a slope of -cot(theta) ?
I wonder if that has to do with the name

Answer 6

sorry, side-thought

Answer 7

so \[\tan \Theta=\sin \Theta/\cos \Theta\] is just an axiom?

Answer 8

An axiom is not the same as a definition. An axiom is a statement we take as an objective fact, but cannot be proved. i.e. a straight line continues infinitely in both directions is an axiom. The concept of the tangent is based on axioms about numbers and geometry, which we have wrapped up and named (defined as) the "tangent function".

Answer 9

tan=sin/cos is not an objective fact, because we didn't have to give a name to that function at all. we could just go around our whole lives writing sin/cos and math would be the same. More annoying, but the same.

Answer 10

Okay then, thank you for your answers.

Answer 11

welcome!