NARRATOR: Exercise 13.
Show that t equals tan of 67.5 degrees
satisfies the quadratic equation t squared minus 2t minus 1 equals 0
and hence find its exact value.
Using the double angle formula for tan,
that is tan of 2x is equal to
2 times tan x divided by 1 minus tan squared x,
we can see that tan of 135 degrees, which is 2 times 67.5 degrees,
is equal to 2 times tan of 67.5 degrees
divided by 1 minus tan of 67.5 degrees all squared.
Substituting the fact that t equals tan of 67.5 degrees,
we see that tan of 135 degrees
is equal to 2t divided by 1 minus t squared.
Now, tan of 135 degrees we can evaluate using the unit circle.
So 135 degrees is an angle in the second quadrant
that is 45 degrees away from the x-axis.
And so we can relate tan of 45 degrees with tan of 135 degrees
using symmetry.
And see that tan of 135 degrees is equal to negative tan of 45 degrees.
Using a "special triangle", we can see that tan of 45 degrees equals 1,
so therefore tan of 135 degrees is equal to negative 1.
So negative 1 equals 2t divided by 1 minus t squared
which means that negative 1 minus t squared is equal to 2t
or t squared minus 1 is equal to 2t
so t squared minus 2t
minus 1 equals 0.
And so we've established that t squared minus 2t minus 1 equals 0.
And so we can go on to solve this equation for t
in order to determine the exact value of tan of 67.5 degrees.
So we first attempt to factorise this quadratic.
Factors of negative 1 that add to negative 2.
And since there are no such factors, we go to the quadratic formula.
So t is equal to 2 plus or minus
the square root of 4 minus 4 times 1 times negative 1 all divided by 2,
which is 2 plus or minus the square root of 4 plus 4 over 2,
which is 2 plus or minus the square root of 8 over 2.
The square root of 8 simplifies to 2 root 2.
And dividing everything through by 2,
we get 1 plus or minus
the square root of 2.
But t, which is tan of 67.5 degrees,
must be positive,
and we know this from the unit circle
that tan of 67.5 degrees
is going to be a positive value.
So therefore tan of 67.5 degrees
must be equal to
1 plus the square root of 2.