Is it possible for tan of 𝜃 to be
equal to negative 125?
To answer this question, we’re
going to recall what we know about the domain and range of the function 𝑓 of 𝜃 is
equal to tan 𝜃. The domain is the set of inputs to
the function that will give a real output. Well, we know that 𝜃 can take all
real values except 90 degrees and then multiples of 180 degrees from there. We could say then that the domain
is the set of real numbers not including 90 plus 180𝑛, where 𝑛 is an integer. In radians, we could say that this
is 𝜋 by two plus 𝑛𝜋 for integer values of 𝑛. Then the range is the output we get
when we input all the values in our domain. Well, as long as we don’t include
𝜋 by two and multiples of 𝜋, our range is all real numbers.
Let’s clarify this by looking at
the graph of the function 𝑓 of 𝜃 is tan 𝜃 or the graph of 𝑦 equals tan of
𝑥. It looks a little something like
this. It’s periodic with a period of 180
degrees. Now, what’s happening at the
asymptotes? Well, as the graph gets closer to
90 plus multiples of 180 degrees, the value of tan of 𝑥 approaches ∞. And so we see our domain is all the
real values but not including the locations of these asymptotes, whereas the range
is the output. It can go up to ∞ and down to
So can it be equal to negative
125? Well, yes, negative 125 is of
course greater than negative ∞ and less than ∞. It’s within our range. And so we say yes, it is possible
for tan of 𝜃 to be equal to negative 125.