Explanation of SOH CAH TOA

Question

I understand that SOH CAH TOA represents a ratio. However, is there any proof or explanation that someone without knowledge on series can understand on how sin cos tan came about? I read my textbook and look on Wikipedia but my knowledge is still practically where I started. Could someone please explain to me the theorem that can be explained to someone whose knowledge is limited to calculus.

Answer 1

Recall that trigonometric functions are defined geometrically and notably

• $\cos x$ and $\sin x$ are the coordinates of the point $M$ on the trigonometric circle
• $\tan x$ is the $y$ coordinate of the intersection between the vertical line from $(1,0)$ and the line $OM$

from here we derive immediately the foundamental relationships $\sin^2 \theta + \cos^2 \theta =1$ and the one you are looking for

$$\tan \theta=\frac{\sin \theta}{\cos \theta}$$

Answer 2

You can view $\text{sine}=\frac {\text{opposite}}{\text{hypotenuse}}$ as the first definition of the sine function and the others similarly. There is no proof of it because it is a definition. The point is that it is a useful definition and the three TLAs are mnemonics for the three definitions.

The reason I say it is a first definition is because it is the one you usually see in a first trigonometry class. Later you may change the definition to depend on the exponential function or the Taylor series. These definitions are equivalent over the reals but easier to extend to complex numbers.