Is the following statement possible or impossible? cos of 𝜃 is equal to negative 3.1.
To answer this question, we’re going to begin by recalling what we know about the domain and range of the function 𝑓 of 𝜃 is equal to cos of 𝜃. The domain is the possible inputs that will give a real output to the function. So in this case, the domain of the function 𝑓 of 𝜃 is equal to cos 𝜃 must be all real values. We can input any real value of 𝜃 into the function 𝑓 of 𝜃 is equal to cos of 𝜃 and get a real output. We call that output the range. So what is the output of our function 𝑓 of 𝜃 is equal to cos of 𝜃?
Well, we know the maximum value of cos of 𝜃 is one, whereas its minimum value is negative one. And so its range is the closed interval from negative one to one. We can see this more clearly if we look at the graph of the function 𝑦 equals sin of 𝑥. It can take any value of 𝑥. The function is just periodic. It will repeat in this manner every 360 degrees. Its output though oscillates between negative one and one inclusive. And so we can clarify that the range is that closed interval from negative one to one.
And this is really useful because we’re asking whether cos of 𝜃 can take the value of negative 3.1. Well, negative 3.1 is considerably less than negative one. It’s outside the range we’ve given. And so this statement is impossible. cos of 𝜃 cannot be equal to negative 3.1 since it can only produce values between and including negative one and one.