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

Is it possible for tan πœƒ to be equal to βˆ’125?

01:49

Video Transcript

Is it possible for tan of πœƒ 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 πœƒ is equal to tan πœƒ. The domain is the set of inputs to the function that will give a real output. Well, we know that πœƒ 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œ‹ by two plus π‘›πœ‹ 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œ‹ by two and multiples of πœ‹, our range is all real numbers.

Let’s clarify this by looking at the graph of the function 𝑓 of πœƒ is tan πœƒ or the graph of 𝑦 equals tan of π‘₯. 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 π‘₯ 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 negative ∞.

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 πœƒ to be equal to negative 125.

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