### Video Transcript

Evaluate two tan 𝜋 over six minus eight sin four 𝜋 over three.

We begin by recalling that 𝜋 radians is equal to 180 degrees. Dividing through by six, this means that 𝜋 over six radians is equal to 180 over six or 30 degrees. In the same way, 𝜋 over three radians is equal to 60 degrees. And multiplying both sides of this by four, we have four 𝜋 over three radians is equal to 240 degrees.

We therefore need to calculate two tan 30 degrees minus eight sin 240 degrees. From our knowledge of special angles, we know that tan of 30 degrees is equal to root three over three. We also know that sin of 60 degrees is equal to root three over two. We can use this together with our knowledge of the CAST diagram to find the sin of 240 degrees. 240 degrees lies in the third quadrant, and the sine of any angle here is negative. We recall that the sin of 180 degrees plus 𝜃 is equal to negative sin 𝜃. This means that the sin of 180 degrees plus 60 degrees is equal to negative sin 60 degrees. sin of 240 degrees is therefore equal to negative sin of 60 degrees, which is equal to negative root three over two.

We can now substitute our values of tan of 30 degrees and sin of 240 degrees into our expression. We have two multiplied by root three over three minus eight multiplied by negative root three over two. This simplifies to two root three over three plus four root three. And since four can be written as 12 over three, this simplifies to 14 root three over three. We can therefore conclude that two tan 𝜋 over six minus eight sin four 𝜋 over three is equal to 14 root three over three.