I know all the regular trig functions like sin = y/r or opp/hyp, cos = x/r or adj/hyp, etc. But what are the inverse trig functions of the 6 trig functions measuring?
Also if anyone can help me with it, could you explain how these inverse functions work on a unit (-1,1) circle instead of how they work a graph from (-inf,+inf)
I’m not sure if this is what you’re looking for, but…
if you use the geometric definitions,
sin (theta) = op / hypotenuse
arcsin (theta) is therefore “the ratio of op : hypotenuse that would generate a right triangle whose opposite angle is theta.” It doesn’t tell you the actual lengths of the sides - an infinite number of possibilities exist - just the ratio of their lengths.
Same for the other inverse trig functions.
Are you asking about arcsin, arctan etc.? These are best thought of as “the angle whose sine is,” “angle whose tangent is” etc.
Xema’s explanation is as good as you’re gonna get for a meaning to those things.
They don’t. The trig functions send a point on the circle to a number. The inverse functions go the other way.
Sure thing. Let me correct your opening statement just ever so slightly though so that we’re speaking in proper terms though. I assume that what you meant to say was “sin (theta) = y/r or opp/hyp, cos (theta) = x/r or adj/hyp” as these functions need an input to produce an output. Specifically, they need an angle as their input, and produce a ratio (that can be thought of as a fraction or a decimal) as their output. These inverse functions, then, can take that ratio and give you back the angle in a right triangle that results in sides having that ratio to each other. So, in other words, we would define them like this: arcsin(opp/hyp) = theta. These inverse functions are very useful for figuring out angles when we know the measures of sides, and also show up in surprising ways in calculus.
Strictly speaking, though, these functions cannot be direct inverses of their counterparts. Sin (x), cos (x), tan (x), and the rest of the crew are all periodic functions, and as you no doubt learned doing inverse functions, a function that is not one-to-one cannot have a direct inverse. We make them inverses by limiting the outputs we look at so that each input in the region has only one output. We also recall that since the range of the cos (x) and sin (x) functions is -1 <= x <= 1, that our domain for their inverses will be -1 <= x <= 1. The tangent function has no such limits on its range (it can output any real number) so we can put in any real number and get an answer for the arctan. It should also be noted that since tan (theta) grows to infinity as you approach pi/2, the arctan of infinity is theta = pi/2.
As for how they relate to the unit circle, well… as you are aware, any point on the unit circle (x, y) can be re-written as (cos (theta), sin (theta)), where theta is the angle formed at the origin by a line through the origin and (x, y) and the x-axis. Similarly, if we are given an x or y coordinate that we are told is on the unit circle, we can find the angle a line through that point makes at the origin through arccos x or arcsin y (depending on which coordinate we are given).
Helpful?
And if you need a calculator for such functions may I suggest one of the most user-friendly is
www.1728.com/trigcalc.htm
It calculates all 6 trig functions and all 6 arcfunctions. It calculates in degrees and radians.
Xema’s explanation is the best.
Without over-explaining this concept here’s yet another example:
The sine of 35 degrees = .57358
The arcsine of .57358 = 35 degrees.