Find the value of sin of seven 𝜋 over six.
We begin by recalling that 𝜋 radians is equal to 180 degrees. Dividing both sides by six, we see that 𝜋 over six radians is equal to 30 degrees. And then multiplying through by seven, seven 𝜋 over six radians is equal to 210 degrees. We therefore need to calculate the sin of 210 degrees.
Recalling our CAST diagram, we see that 210 degrees lies in the third quadrant, and the sine of any angle here is negative. We can go one stage further using our properties of related angles. The sin of 180 degrees plus 𝜃 is equal to negative sin 𝜃. This means that sin of 180 degrees plus 30 degrees is equal to negative sin of 30 degrees. sin of 210 degrees is therefore equal to negative sin of 30 degrees.
We know that 30 degrees is one of our special angles and that the sin of 30 degrees is one-half. The sin of 210 degrees is therefore equal to negative one-half. And we can therefore conclude that the sin of seven 𝜋 over six radians is equal to negative one-half.