How this instrument works
The three inverse trigonometric functions — arcsine, arccosine, and arctangent — each undo a different ordinary trig function, and this page runs all three at once on the same input so their differing behavior is visible directly. Given a value x between −1 and 1 (the range where asin and acos both stay valid), it returns asin(x), acos(x), and atan(x) together.
A fixed relationship links two of the three at every valid input: asin(x) + acos(x) always equals exactly 90°, since sine and cosine are complementary functions. atan(x), by contrast, follows no such fixed relationship with the other two — it's built from a genuinely different ratio (opposite over adjacent, rather than opposite or adjacent over the hypotenuse), so its value at a shared input doesn't line up with the other two in any similarly clean way.
This comparison view is useful specifically for building intuition about how the three functions relate — a student meeting inverse trig for the first time can watch, say, x = 0.5 send asin to 30°, acos to 60°, and atan to a genuinely unrelated 26.57°, making the distinction concrete rather than abstract.
- Enter a value between −1 and 1 into the x field.
- Read asin(x), acos(x), and atan(x) together: all three inverse trig results for that same input.
- Watch how asin and acos always sum to 90°, while atan follows its own independent path.
Worked example — all three at x = 0.5
At x = 0.5, the three functions give genuinely different results: asin(0.5) = 30°, acos(0.5) = 60°, and atan(0.5) ≈ 26.57°. Note asin and acos sum to exactly 90° (30+60), confirming their complementary relationship, while atan's result doesn't fit that same pattern at all.
At x = 0, asin(0) = 0° and atan(0) = 0° happen to agree, while acos(0) = 90° sits at its own separate reference point. At x = 1, the roles flip: asin(1) = 90° and acos(1) = 0° have swapped entirely, while atan(1) lands on its own clean 45°, unrelated to either.
Questions
Why compare all three inverse trig functions on one page?
To make their differing behavior concrete rather than abstract — seeing asin, acos, and atan respond differently to the identical input, side by side, builds intuition faster than checking three separate dedicated pages one at a time.
Why do asin and acos always sum to 90°?
Because sine and cosine are complementary functions: sin(θ) = cos(90°−θ) for any angle θ. This identity carries straight through to their inverses, so asin(x) + acos(x) = 90° holds for every valid shared input x.
Why doesn't atan follow that same 90° pattern?
Because tangent is built from a fundamentally different ratio than sine or cosine — opposite over adjacent, rather than either side over the hypotenuse — so its inverse doesn't share the same complementary relationship the other two enjoy with each other.
Why is the input restricted to −1 through 1 here, even though atan alone accepts any real number?
Because asin and acos are only defined within that range, and this page compares all three at a shared input — restricting to the range all three can handle keeps the comparison meaningful across every entry.
How is this different from the individual asin, acos, and atan pages?
Those pages each show one function's result on its own. This page runs all three simultaneously on the identical input, specifically to make the relationships and differences between them visible at a glance.