Why is tan90⁰ not defined?

In the series of our posts after our last post titled How to prepare for 10th boards, 12th boards, IIT and NEET Exams We at Alchemy Learning bring you a very basic but valuable concept which can bring a change in the way you prepare for your exams

Trigonometric ratios are defined as the ratios of sides of a right angle triangle till class 10^th but trigonometric ratios are not constrained to a right angle triangle. In earlier classes tan𝛉 is defined as the ratio of perpendicular to base in a right angle triangle and tan90^o is written as ∞ in trigonometric tables.

We see in trigonometric table there is one value of tan𝛉 written as “NOT DEFINED”. But we never ask ourselves WHY?

Considering 𝛉 = 0^o

Let us consider a right angle triangle ABC which has right angle (90^o) at B and consider angle A = 𝛉, AC = hypotenuse, BC = perpendicular and AB = base.

When we consider values of 𝛉 to be 30^o, 45^o and 60^o we can see it is possible in a right angle triangle. But what about 0^o and 90^o. If 𝛉 is 0^o this means hypotenuse AC is overlapping base AB and they are of same length. This implies ABC is no longer a triangle instead they are a pair of straight lines exactly overlapping each other. This conclusion tells us that our perpendicular is zero, hence sin𝛉 and tan𝛉 both are zero and cos𝛉 is 1.

Considering 𝛉 = 90^o and Defining the Problem

Considering 𝛉 = 0^o is simpler to understand but assuming 𝛉 = 90^o is very difficult to understand. Let’s try to figure out why is it a puzzle and then we will try to solve that puzzle.

When we consider 𝛉 = 90^o we will have two 90^o angles in a triangle which is not possible. Two 90^o angles implies that hypotenuse AC and perpendicular BC both are parallel to each other and they will never meet. So, it is impossible to tell what is the length of perpendicular and hypotenuse. Some of you might think that they will meet at infinity (very far away) and perpendicular and hypotenuse both of them will be ∞ . But this consideration is wrong let’s find out how?

CASE-I

𝛉 slightly less than 90^o

In this case ABC will be a right angle triangle with hypotenuse and perpendicular having almost equal lengths and base is very-very small compared to both hypotenuse and perpendicular. AC and BC will meet at C point above the line AB. So tan𝛉 will be +∞ as perpendicular >>> base.

CASE-II

𝛉 slightly greater than 90^o

In this case ABC will be a right angle triangle but it will be inverted. Again this time also hypotenuse and perpendicular will be of almost equal lengths, very-very small base but AC and BC will meet at C point below the line AB. C point is below the reference line AB, so perpendicular will be negative. So tan𝛉 will be -∞ as perpendicular >>> base in magnitude and perpendicular is taken as negative which results in negative sign in value of tan𝛉.

Now, in first case where 𝛉 is slightly less than 90^o tan𝛉 is +∞ and in second case where 𝛉 slightly greater than 90^o tan𝛉 is -∞. For a very small change (almost negligible) in 𝛉, tan𝛉 changes its value from -∞ to +∞. So we can say that at 𝛉 = 90^o tan𝛉 will have a value between -∞ to +∞. The only problem here is that we cannot point which value tan𝛉 will have. It can be anything on the number line. This is why we say tan𝛉 is not defined at infinity.

Graphical Method

By looking at the graph you can understand that as angle approaches ℼ/2 or 90^o from left side tan𝛉 approaches +∞ and as angle approaches ℼ/2 from right side tan𝛉 approaches -∞. So we cannot say what will be the value of tan𝛉 at 90^o. Hence it is written as not defined at 𝛉 = 90^o.

NOTE: To understand the concepts better it is very important that we ask questions about everything which is being taught to us in our schools or coaching. It is not only about this particular question but for every question that is left unanswered and which leads to a confusion later on. Students preparing for JEE MAINS, JEE ADVANCED (IIT), NEET, AIIMS, BITSAT and etc. it is a very important lesson and they should imbibe this nature of asking questions in their life.

Clearing this kind of concepts is the key to success for students aspiring NTSE, KVPY, Olympiad as it becomes the basic foundation on which Maths’ and Physics’ success is built. In the case you have any query regarding this concept you are welcome  to interact with our experts  either on our facebook page or on AlchemyLearning’s website.