There is a right triangle with a left side of six, a bottom side of 10, and a hypoteneuse 11.66. How do you find the other two angles? I tried the inverse of tangent 6/11.66, but my calculator said that's invalid. What do you properly do?

You probably omitted some parentheses or pressed the wrong keys.

The inverse tangent of 6/11.66 exists, but does not represent the angle you are looking for.
Lets go back to the definition of the tangent, which is
tangent(θ)=opposite/adjacent.
If we consider the right-angle to be on the left, and the angle θ on the right of the bottom side of 10, then
opposite = 6,
adjacent = 10
θ = arctangent(6/10) = approx. 31°
The top angle can be obtained by subtraction,90°-θ

also tangent is opposite over adjacent

To find the other two angles of a right triangle, you can use the trigonometric functions. Let's start with finding the angle opposite the left side of length 6.

You correctly attempted to use the inverse tangent (arctan) function, but there seems to be a misunderstanding about the inputs. The tangent function relates the ratio of the lengths of the sides of a right triangle with one of the acute angles, not the inverse.

To find the angle opposite the left side (let's call it angle A), you can use the inverse sine function (arcsin) instead. The formula is:

sin(A) = length of opposite side / length of hypotenuse.

In this case, sin(A) = 6 / 11.66. To find angle A, you can take the inverse sine of this value:

A = arcsin(6 / 11.66).

Now, let's move on to finding the other angle (angle B) opposite the bottom side of length 10. Since we know that the sum of angles in a triangle is 180 degrees, we can determine angle B by subtracting angle A from 90 degrees:

B = 90 degrees - A.

To find angle B, you can substitute the value of angle A that we calculated previously into this equation.

Once you have both angle A and angle B, you can determine the third angle (angle C) by realizing that it is the right angle and therefore equal to 90 degrees.

Remember to use your calculator's inverse sine function (often written as sin^(-1) or asin) to find the value of arcsin(6 / 11.66) accurately.