A triangle consists of three sides with three angles between each combination of two sides in the triangle. The three angles add up to 180 degrees, or “pi” radians (depending on the unit chosen). If one of the three angles is 90 degrees, the triangle is called a right triangle. For one of the non-ninety degree angles (call it angle “X”), the cosine of that angle is equal to the length of the non-hypotenuse side adjacent to the angle divided by the length of the hypotenuse. Calling the ratio of those two sides, Y, we have
cos(X) = Y
Calculating Arccos from Cos
Arccos is short for arccosine. It is a kind of inversion, but not according to the usual sense of the word inversion. It is incorrect to say,
arccos(X) = 1/cos(X) = 1/Y
Rather, the inverse is of a different form. It is like removing the cosine function. Thus,
If cos(X) = Y,
Then, taking the arccos of both sides of the equation doesn’t take away its equality, so,
arccos(cos(X)) = arccos(Y),
X = arccos(Y).
This is because writing arccos(cos(X)) is the same thing as writing X.
Thus arccos is the inverse of the functionality of cos, not the inverse of the numerical value of cos.
Confused? Consider an example. The cosine or “cos” of a 45 degree angle is (sqrt 2)/2.
cos(45) = (sqrt 2)/2
The arccosine is,
45 = arccos((sqrt 2)/2)
Another way to simplify this in our heads is to say,
“45 is the number whose cosine is the square root of two over two.”
Do you understand the concept of cosine, but still don’t get it? In that case, more than one reading of this article in its entirety will likely impart the understanding you seek.
References and Resources:
Math is Fun – Sine, Cosine, and Tangent
University of Cambridge – Math Thesaurus