# Explanation of SOH CAH TOA

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.

• What is SOA CAH TOA? – user Sep 18 '18 at 21:02
• SOH CAH TOA is a mnemonic device to remember that the geometric interpretations of sine, cosine, and tangent as as the ratios $\dfrac{\text{opposite}}{\text{hypotenuse}},\dfrac{\text{adjacent}}{\text{hypotenuse}},\dfrac{\text{opposite}}{\text{adjacent}}$ respectively. That is to say, for example, that given a right triangle and looking at a particular corner (other than the corner with the right angle), the sine of that angle is the length of the edge opposite to our corner divided by the length of the hypotenuse. – JMoravitz Sep 18 '18 at 21:07
• @JMoravitz Thanks! I didn't know that mnemonic device :) – user Sep 18 '18 at 21:13
• Given a right triangle with hypotenuse $1$, if we know that one of the triangle's angles is $\theta$, it's natural to ask what are the lengths of the adjacent and opposite sides of the triangle. That is just a very basic question, something we will inevitably want to know. The answers are named $\cos(\theta)$ and $\sin(\theta)$. You might object that giving names to these lengths does not tell us what the lengths are. But, naming the unknowns is a good first step towards figuring out how to compute their values. – littleO Sep 18 '18 at 21:14

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

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

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.