True or False: If sin 𝜃 is equal to three-fifths and cos 𝜃 is less than zero, then tan 𝜃 is equal to negative three-quarters.
We begin by sketching the CAST diagram for angles between zero and two 𝜋 radians. We know that in the first quadrant, the sine, cosine, and tangent of any angle are all positive. In the second quadrant, the sin of angle 𝜃 is positive, whereas the cos and tan of angle 𝜃 are negative. In the third quadrant, only the tangent function is positive, and in the fourth quadrant, only the cosine function is positive.
In this question, we are told that sin 𝜃 is equal to three-fifths and is therefore positive. The cos of angle 𝜃 is negative. For both of these to be true, we know that 𝜃 must lie in the second quadrant. And this means that the tan of angle 𝜃 must be negative. Sketching a right triangle in the second quadrant, we see that this is a Pythagorean triple consisting of the three positive integers three, four, and five such that three squared plus four squared is equal to five squared.
Since the tangent of any angle in a right triangle is equal to the opposite over the adjacent, the tan of angle 𝛼 is equal to three-quarters. We see from the diagram that 𝜃 is equal to 180 degrees minus 𝛼. And we recall that the tan of 180 degrees minus 𝛼 is equal to negative tan 𝛼. The tan of angle 𝜃 is therefore equal to negative three-quarters.
And we can conclude that the statement is true. If sin 𝜃 is equal to three-fifths and cos 𝜃 is less than zero, then tan 𝜃 is equal to negative three-quarters.