Video Transcript
Is sin 30 degrees positive or negative?
In order to answer this question, we will begin by splitting the 𝑥𝑦-coordinate plane into four quadrants. We measure any positive angle in a counterclockwise direction from the positive 𝑥-axis. This means that quadrant one contains angles between zero and 90 degrees, quadrant two between 90 and 180 degrees, and so on. The angle 30 degrees lies in the first quadrant, as shown.
One way of recalling whether the trigonometric functions sine, cosine, and tangent are positive or negative in each quadrant is using the CAST acronym. In quadrant one, the A represents all. If any angle 𝜃 lies between zero and 90 degrees, then the sin of angle 𝜃, cos of angle 𝜃, and tan of angle 𝜃 are all positive. We can therefore conclude that the sin of 30 degrees is positive. 30 degrees is actually one of our special angles. And we know that the sin of 30 degrees is one-half. This confirms the correct answer of positive.
Whilst it is not required in this question, it is worth recalling what the letters S, T, and C represent on the CAST diagram. In the second quadrant, the S tells us that the sin of any angle 𝜃 between 90 and 180 degrees is positive, whereas the cosine and tangent of any angle in this quadrant is negative. The T in quadrant three tells us that the tangent of any angle between 180 and 270 degrees is positive. The sine and cosine of any angle in this quadrant is negative. Finally, in quadrant four, the cosine of any angle is positive, whereas the sine and tangent of any angle between 270 and 360 degrees is negative. This leads us to the fact that each of the three trigonometric functions are positive in two quadrants and negative in the other two.
