Video Transcript
If cot 𝜃 is equal to negative four-thirds and cos 𝜃 is greater than zero, find csc 𝜃.
We begin by recalling one of the reciprocal trigonometric identities, which states that cot 𝜃 is equal to one over tan 𝜃. In this question, we are told that cot 𝜃 is equal to negative four-thirds. tan 𝜃 will be the reciprocal of this, so is therefore equal to negative three-quarters. Next, we will use our knowledge of the CAST diagram to find the quadrant that 𝜃 lies in. We are told that cos 𝜃 is positive, and we have just worked out that tan 𝜃 is negative. This only occurs in the fourth quadrant, where the cos of any angle is positive, whereas the sin and tan of any angle are negative.
We are asked to find the csc of angle 𝜃, and we know this is the reciprocal of sin 𝜃. Using the fact that the tan of 𝜃 is equal to negative three-quarters, we can sketch a right triangle in the fourth quadrant. This triangle is a Pythagorean triple consisting of three positive integers three, four, and five such that three squared plus four squared is equal to five squared. Since the sin of angle 𝛼 is equal to the opposite over the hypotenuse, from the diagram we see that sin 𝛼 is equal to three-fifths. We can also see from the diagram that 𝜃 is equal to 360 degrees minus 𝛼.
And using the properties of related angles, we know that the sin of 360 degrees minus 𝛼 is equal to negative sin 𝛼. This means that sin 𝜃 is equal to negative sin 𝛼, which is equal to negative three-fifths. As already mentioned, csc 𝜃 is equal to one over sin 𝜃. So csc 𝜃 is equal to negative five-thirds.
We can therefore conclude that if cot 𝜃 is equal to negative four-thirds and cos 𝜃 is greater than zero, then csc 𝜃 is equal to negative five-thirds.
