00:01
We are going to find all solutions of the equation sine of x plus pi minus sine of x plus square root of 2 equals 0 in the interval 0 to pi, including 0 but not including 2 pi.
00:13
So first, the first thing we got to do here in this equation is simplify the term sine of x plus pi, because we have the sine of the sum of two angles.
00:23
And we note that's equal to sine of the first angle sine of x times cosine of the second angle pi plus sine of the second angle pi times cosine of the first angle x and now we know sine of of pi is 0 and cosine of pi is negative 1.
00:50
So this is equal to negative sine of x.
00:55
So we can say that sine of x plus pi minus sine of x plus square root of 2 equals 0 is equivalent to negative sine of x.
01:17
So here minus sine of x plus square root of 2 equals 0 which in its turn is equivalent to negative 2 sine of x plus square root of 2 equals 0 and this is equivalent to 2 sine of x equals square root of 2 and finally this is the same as sine of x equal square root of 2 over 2 so the equation we want to solve here is just the same as this one so the solutions are the same so we solve this equation here when or for what values of x sine of x is square root of 2 over 2 square root of 2 over 2 is the same as 1 over square root of 2 is a number a positive number and less than one so it makes sense because sine of x only takes values between negative one and one so if we draw if we sketch the function sine of x on the interval zero to pi we get something like this more or less maximum value being one here at pi half 1 pi half, pi half, this is pi, this is 3 pi half, where the value is 81, and here we have 2 pi.
03:03
So the value 1 over square root of 2, or what is the same square root of 2 over 2, is a number, let's say 0 .7, it's a number somewhere around here...