Proving the Validity of Sin, Cos, and Tan Equations in Right Triangles

[mather]

hello!

we know that in every right triangle there are the sin, cos, tan etc equations

how do we prove that these equations are valid?

eg. how do we prove that the adjacent of an angle divided by the hypotenuse of the triangle is always the same for that given angle?

thanks

 

[Mark44]

hello!

we know that in every right triangle there are the sin, cos, tan etc equations

how do we prove that these equations are valid?

We don't. These are definitions. Definitions are never proved.

eg. how do we prove that the adjacent of an angle divided by the hypotenuse of the triangle is always the same for that given angle?

thanks

 

[mather]

how do we define something if we can't prove it is correct?

 

[Mark44]

You can define something any way you want. Whether it is correct or not is another matter. For a right triangle, sin(θ) = opposite/hypotenuse, where θ is one of the two acute angles.

Definitions in mathematics work in about the same way as definitions of words in a language. If you look up the word "cat", the dictionary gives you the definition; i.e., tells you what the word "cat" means.

 

[micromass]

hello!

we know that in every right triangle there are the sin, cos, tan etc equations

how do we prove that these equations are valid?

How did you define sine, cosine and tangent functions? The answer will depend on this.

eg. how do we prove that the adjacent of an angle divided by the hypotenuse of the triangle is always the same for that given angle?

You will need the concept of similar triangles. Are you familiar with this?

 

[micromass]

You can define something any way you want. Whether it is correct or not is another matter. For a right triangle, sin(θ) = opposite/hypotenuse, where θ is one of the two acute angles.

Definitions in mathematics work in about the same way as definitions of words in a language. If you look up the word "cat", the dictionary gives you the definition; i.e., tells you what the word "cat" means.

I think the poster wants to know why our definition of sine as

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypothenuse}}$$

is independent of the triangle. Since given two arbitrary rectangular triangles, it is certainly possible that the opposite sides and the hypothenuse are completely different. In order for our definition of the sine to be a good one, we need the quotients to equal for every triangle which possesses an angle $\theta$.

 

[Mark44]

I think the poster wants to know why our definition of sine as

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypothenuse}}$$

is independent of the triangle. Since given two arbitrary rectangular triangles, it is certainly possible that the opposite sides and the hypothenuse are completely different. In order for our definition of the sine to be a good one, we need the quotients to equal for every triangle which possesses an angle $\theta$.

Which we can prove by similar triangles in plain old geometry, as you said. The OP's question wasn't clear to me.

 

[mather]

that similar triangles in plain old geometry seems to be what I need

 

[mather]

that similar triangles in plain old geometry seems to be what I need

anyone?

 

[Mark44]

that similar triangles in plain old geometry seems to be what I need

anyone?

Yes. No one responded because they probably didn't think you still had a question. A question ends with a ?.

 

[mather]

Which we can prove by similar triangles in plain old geometry, as you said.

how do we actually prove that?

 

[micromass]

how do we actually prove that?

Do you know Thales' theorem on similar triangles?

 

[mather]

nope

 

[micromass]

nope

So what Euclidean geometry do you know? What do you know about triangles? What do you know about similarity?

 

[Borek]

nope

Time to ask uncle google. micromass is pretty busy with other things and while he can try to spoonfeed you, you risk getting the spoon content all over you sooner or later.