So we know about sine, cosine, and tangent... and we even know about cosecant, secant, and cotangent, their multiplicative inverses. So these inverse functions are the compositional inverse. I'm talking about arcsin, arccosine, and arctangent - but if you've seen a graphing calculator, you'd probably recognize them by their more common notation: sin^-1, cos^-1, and tan^-1. Don't let the negative one exponent fool you - this is simply how we denote that we're looking for the inverse function. So what is that anyway? Well, look at the triangle here.
Say we know the length of sides a and b but we don't know the value of the angle θ.
Given what we already know, we can clearly say that tanθ = a / b, but that doesn't really tell us what the value of θ is. That's where the inverse functions come in. See, where the tanθ = a / b, the arctan a / b = θ! The same is true for the sine, cosine and all the rest. Simply put, when you're trying to get an angle and you know the values of some sides, you're going to be looking for inverse functions. Plus - they can be graphed just like our other functions can, but the results may surprise you. Take a look at what happens when we graph arcsin(x):
Now - think about this for a second and you may realize why this is the case. Picture the value of arcsin(x) as the angle at the center of the unit circle (because that's what it is). That value ranges, in each "quadrant" of the graph, between π/2 radians and - π/2 radians - or, 90 degrees and -90 degrees. Now picture those values rolled out as you continue around the circle. Your angle may be 5π radians, but you're in the same place as if you had only gone π radians - remember, one trip around the circle is 2π, so reduce in terms of that. Also, see that the value of sin(π) is the same as the value of sin(0). They're both zero.
In order to keep this from happening with inverse trig functions, and to actually make them functions, as the name implies, we have to define ranges for them. If we define the range of arcsin(x) as [- π/x, π/2] and the domain as [-1,1] we get something much nicer:
That's all for now - more to come very soon, promise!