+5478 While the cosine of a given angle is used to find the ratio of its opposite side / hypotenuse, the arccosine does the opposite: it finds the measure of an angle from the given ratio of its opposite side / hypotenuse.
For example, $$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{cos}}{\left({\mathtt{45}}^\circ\right)} = {\frac{{\sqrt{{\mathtt{2}}}}}{{\mathtt{2}}}}$$
So $$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{cos}}^{\!\!\mathtt{-1}}{\left({\frac{{\sqrt{{\mathtt{2}}}}}{{\mathtt{2}}}}\right)} = {\mathtt{45}}$$
So if you had the lengths of the opposite side and the hypotenuse for a right triangle, you would use arccosine to find the measure of the angle.
+5478 While the cosine of a given angle is used to find the ratio of its opposite side / hypotenuse, the arccosine does the opposite: it finds the measure of an angle from the given ratio of its opposite side / hypotenuse.
For example, $$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{cos}}{\left({\mathtt{45}}^\circ\right)} = {\frac{{\sqrt{{\mathtt{2}}}}}{{\mathtt{2}}}}$$
So $$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{cos}}^{\!\!\mathtt{-1}}{\left({\frac{{\sqrt{{\mathtt{2}}}}}{{\mathtt{2}}}}\right)} = {\mathtt{45}}$$
So if you had the lengths of the opposite side and the hypotenuse for a right triangle, you would use arccosine to find the measure of the angle.
