Question Video: Determining Whether a Value Is in the Range of a Trigonometric Function

Video Transcript

Is the statement sec 𝜃 equals 0.9 possible or impossible?

To answer this question, we’re going to begin by recalling what we know about sec of 𝜃. sec of 𝜃 is equal to one over cos of 𝜃. So, what we’re really asking here is, is the statement one over cos 𝜃 equals 0.9 possible or impossible? Now, we actually know quite a lot about the function cos of 𝜃. So, we’re going to rearrange to make cos of 𝜃 the subject. We’ll begin by multiplying both sides of this equation by cos 𝜃, giving us one equals 0.9 cos 𝜃. Then, we’ll divide through by 0.9. So, cos 𝜃 is one divided by 0.9 or cos of 𝜃 is equal to ten-ninths. This is the statement that we’re looking to decide is possible or impossible.

And so next, we look at the domain and range of the function cos of 𝜃. The domain is all possible inputs to the function that will give a real output. Well, 𝜃 can take all real numbers and will still give an output for cos of 𝜃. So, the domain of the function 𝑓 of 𝜃 equals cos of 𝜃 is all real numbers. But what about the range of this function? Well, one way to decide is to think about the graph of 𝑦 equals cos of 𝑥. It moves between one and negative one in the vertical direction. So, its output never exceeds one or negative one.

And this means the range of the function 𝑓 of 𝜃 is cos 𝜃 must be the closed interval from negative one to one. In other words, for real values of 𝜃, cos of 𝜃 will always be greater than or equal to negative one and less than or equal to one. Remember though, we’re looking to decide whether cos of 𝜃 can be equal to ten-ninths. But ten-ninths is greater than one. So, we say that cos of 𝜃 cannot be equal to ten-ninths, meaning that sec of 𝜃 cannot be equal to 0.9. And so, we see that the statement sec of 𝜃 is 0.9 is impossible.