True or False: Each of the trigonometric functions is positive in only one quadrant.
We begin by recalling that the four quadrants in the 𝑥𝑦-plane are as shown. We measure positive angles in a counterclockwise direction from the positive 𝑥-axis. This means that the first quadrant contains angles between zero and 90 degrees, the second quadrant between 90 and 180 degrees, and so on.
One way of recalling whether the trigonometric functions are positive or negative in each quadrant is using the CAST acronym. The letter A in the first quadrant stands for all, as all three of sin 𝜃, cos 𝜃, and tan 𝜃 are positive when the angle 𝜃 lies between zero and 90 degrees. The sine of any angle in the second quadrant between 90 and 180 degrees is also positive. However, the cosine and tangent of any angle in this quadrant is negative. In the third quadrant, the tangent of any angle is positive, whereas the sine and cosine of any angle between 180 and 270 degrees is negative. Finally, in the fourth quadrant, the cosine of any angle is positive, whereas the sine and tangent of any angle is negative.
We can therefore conclude that the sine function is positive in the first and second quadrants. The cosine function is positive in the first and fourth quadrants. And the tangent function is positive in the first and third quadrants. Summarizing this, we see that all three trigonometric functions are positive in two quadrants. This means that the initial statement is false.