# Prove sin, cos, tan

1. Apr 19, 2014

### 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

2. Apr 19, 2014

### Staff: Mentor

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

3. Apr 19, 2014

### mather

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

4. Apr 19, 2014

### Staff: Mentor

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.

5. Apr 19, 2014

### micromass

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

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

6. Apr 19, 2014

### micromass

Staff Emeritus
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 $\theta$.

7. Apr 19, 2014

### Staff: Mentor

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

8. Apr 19, 2014

### mather

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

9. Apr 21, 2014

### mather

anyone???

10. Apr 21, 2014

### Staff: Mentor

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

11. Apr 21, 2014

### mather

how do we actually prove that?

12. Apr 21, 2014

### micromass

Staff Emeritus
Do you know Thales' theorem on similar triangles?

13. Apr 21, 2014

### mather

nope

14. Apr 21, 2014

### micromass

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

15. Apr 21, 2014

### Staff: Mentor

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.