True or false: If sin 𝑥 is greater than zero, then the angle must only be in the first quadrant.
We begin by recalling that the four quadrants on the 𝑥𝑦-coordinate plane are as shown. Positive angles are measured in the counterclockwise direction from the positive 𝑥-axis. And we can therefore add zero, 90, 180, 270, and 360 degrees, as shown. When dealing with sine, cosine, and tangent ratios, we can add the letters A, S, T, and C to the diagram. This is often referred to as a CAST diagram.
In the first quadrant, where our angle 𝑥 lies between zero and 90 degrees, the sin of angle 𝑥, cos of angle 𝑥, and tan of angle 𝑥 are all positive. In the second quadrant, when the angle lies between 90 and 180 degrees, only the sin of the angle is positive. In this quadrant, the values of cos 𝑥 and tan 𝑥 are both negative. In the third quadrant, the letter T tells us that the tan of any angle between 180 and 270 degrees is positive. sin 𝑥 and cos 𝑥 are both negative in the third quadrant. Finally, the cosine of any angle between 270 and 360 degrees is positive, whereas the sine and tangent of any angle in this quadrant is negative.
Whilst we could prove these using the unit circle or the graphs of our trigonometric functions, it is enough to simply quote the CAST diagram. We can therefore conclude that sin 𝑥 is greater than zero in the first and second quadrants. When the angle 𝑥 lies between zero and 180 degrees, then sin 𝑥 is positive. The statement “If sin 𝑥 is greater than zero, then the angle must only be in the first quadrant.” is false.