00:01
So in this problem, we want to solve this trigonometric expression exactly without the use of a calculator.
00:09
So we have the sine of the quantity, arc sine of 5 over 6 plus arc cosine of negative 4 over 9.
00:18
Okay, so let's start with this first piece, right? i have the arc sine of 5 over 6 means i have a triangle here.
00:28
I have a right triangle.
00:29
Well, i didn't draw that very well.
00:30
Of the right triangle like such, where this is my theta, is such that the sign of it is 5 over 6, which means this side's 5, this side's 6, and so this side is by pythagorean theorem, 36 minus 25 is 11, right? the square of the, some of the square of the legs is the square of the hypotenuse.
01:04
So i can take the square of the hypotenuse minus that one leg know and take the square root of that result so that's the square root of 11 okay so then that means that the cosine of theta here is the square root of 11 over 6 where the sign of theta was given to us right essentially as 5 over 6 okay so let's look at the other one cosine inverse of negative 4 over 9 okay, so that means then that this right triangle goes like this, doesn't it? where i have, here is alpha, angle alpha, and the hypotenuse is 9, the cosine of this, this would be negative 4 then, right? so i get the cosine of it.
02:07
And so then by pythagorean theorem, 9 squared is 81, minus 4 squared is 16, and that gives me 65.
02:21
And so this side is the square root of 65...