Text description - screencast for exercise 13

[The narrator reads out the onscreen text.]

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.