True or False: The cosine function is positive in the second and fourth quadrants.
We begin by recalling that the four quadrants on the 𝑥𝑦-coordinate plane are labeled as shown. We measure positive angles from the positive 𝑥-axis in the counterclockwise direction. The angles zero, 90, 180, 270, and 360 degrees divide our quadrants.
One way of recalling whether the sine, cosine, and tangent functions are positive or negative in each quadrant is using the CAST acronym. In the first quadrant, the A stands for all, where the sin, cos, and tan of any angle 𝜃 between zero and 90 degrees are all positive. The S in quadrant two tells us that the sin of any angle 𝜃 between 90 and 180 degrees is positive. The cosine and tangent of any angle in this quadrant are negative. In quadrant three, the tangent of any angle is positive, whereas the sine or cosine of any angle between 180 and 270 degrees is negative. 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.
In this question, we are interested in the quadrants where the cosine function is positive. From the CAST diagram, we see that the cosine function is positive in the first and fourth quadrants. This means that the initial statement is false as the cosine function is not positive in the second quadrant.