Are there 3 trig functions or are there 6 trig functions? In my algebra class we are being taught that there are only the 3 basic trig functions (cosine, sine, and tangent). But my friend who is 2 math grade levels ahead of me is saying that there is 6 trig functions (cosine, sine, tangent, cotangent, secant, and cosecant). Does anyone know why we are being taught differently and which one of us is correct?
It depends on how you look at it I guess, but:
$$\cot(x) = \frac{1}{\tan(x)}$$
$$\csc(x) = \frac{1}{\sin(x)}$$
$$\sec(x) = \frac{1}{\cos(x)}$$
So the three &quot;extra&quot; functions your friend told you about are just derived from the three you know. But if that's the rule, then two of the ones you know,
$$\cos(x) = \sin\left(\frac{\pi}{2} - x\right)$$
$$\tan(x) = \frac{\sin(x)}{\cos(x)} = \frac{\sin{x}}{\sin\left(\frac{\pi}{2} - x\right)}$$
are also just derived functions. Hence we would say there is only one trigonometric function, for example $\sin{x}$.
(As others have mentioned, this statement works even counting hyperbolic functions, because of properties like $\cosh(x) = \cos(ix)$ and so on, or using $e^{i\theta} = \cos{\theta} + i\sin{\theta}$. But since you don't appear to be at this level of math yet, I won't go into detail about that.)
Bottom line: We only need one trigonometric function, but for practical reasons, there are more.
Actually there is hyperbolic, elliptical, cardioid, integral, cubic, quartic, and many more. It can be viewed as curve functions.