Find sin of negative 690 degrees without using a calculator.
In order to answer this question, we will use our knowledge of special angles together with the CAST diagram. We begin by recalling that negative angles, as in this question, are measured in a clockwise direction from the positive 𝑥-axis. One full turn in this direction takes us to negative 360 degrees. Our angle is more negative than this, so we will consider a second turn. We see that the angle negative 690 degrees is as shown, and this lies in the first quadrant. This angle is 30 degrees away from the positive 𝑥-axis. And this means that the sin of negative 690 degrees is equal to the sin of 30 degrees. Negative 690 degrees and 30 degrees are coterminal angles, as they have the same terminal side in standard position.
The angle 30 degrees is one of our special angles. And we know that the sin of 30 degrees is equal to one-half. This means that the sin of negative 690 degrees is also equal to one-half. And we have therefore worked out the angle required without using a calculator.