The value of tan 90 degrees is not defined. In Trigonometry, Sine, Cosine and Tangent are the three primary ratios, based on which the whole trigonometric functions and formulas are designed. Each trigonometric functions have their equivalent importance. Basically, these ratios are used to find the angles and the sides of a right-angled triangle. These angles are calculated with respect to sin, cos and tan functions.

Usually, the degrees are considered as 0°, 30°, 45°, 60°, 90°, 180°, 270° and 360°. In this article, we will discuss, how to find the value for tan 90 degrees along with other degrees or radians.

Value of Tan 90 Degrees

The exact value of tan 90° is:

Note: Tan 90 = Cot 0 = ∞

How to Find Value of Tan 90

As discussed, when we speak about trigonometry, Sine, Cosine and Tangent are the principle trigonometric functions. Let us give a brief about all three functions or ratios with respect to a right-angled triangle.

Sine functions denote that for a given right-angled triangle, the sin of angle θ is equal to the ratio of the opposite side to the angle, and hypotenuse.

Sin θ =Opposite Side/Hypotenuse

Cosine function denotes that for a given right-angled triangle, the cos of angle θ is equal to the ratio of the adjacent side to the angle, and hypotenuse.

Cos θ =Adjacent Side/Hypotenuse

Tangent function denotes that for a given right-angled triangle, the tan of angle θ is equal to the ratio of the opposite side to the angle, and adjacent side or base.

Tan θ = Opposite Side/Adjacent Side

We can also represent the tangent function as the ratio of the sine function and cosine function.

∴ Tan θ = Sin θ /Cos θ

So, tan 90 degrees in terms of ratio is,

Tan 90° = Sin 90⁰ / Cos90°

From the trigonometric table, we know,

Sin 90° = 1

And,

Cos 90° = 0

∴ Tan 90°= 1/0 = Undefined

That means, we cannot define Tan 90⁰ value.

For a unit circle, which has a radius equal to 1, we can derive the tangent values of all the degrees. With the help of a unit circle drawn on the XY plane, we can find out all the trigonometric ratios and values. As you can see in the graph, Tan 90 degrees unit circle value is undefined or infinite.

In the same way, we can derive other values of tangent degrees (0°, 30°, 45°, 60°, 90°, 180°, 270° and 360°). Below is the trigonometry table, which defines all the values of tangent along with other trigonometric ratios.

Why is Tan 90 undefined?

Tangent 90 degrees is evaluated as undefined because tan of an angle is equal to the ratio of sin and cos of same angle. Since, sin 90 = 1 and cos 90 = 0, therefore;
Tan 90 = sin 90/cos 90
= 1/0
= ∞
As we have got the result as infinity, and we cannot define infinity, therefore tan 90 is undefined.

Trigonometry Equations Based on Tangent Function

Tangent functions are used to formulate multiple trigonometric formulas.

The basic formula for the tangent function is;

Tan θ = Perpendicular/ Base

Alternatively,

Tan θ = sin θ/cos θ

Or

Tan θ = 1/Cot θ

Other formulas:

1. Tan(-θ )=-Tanθ
2. Tan (x+y)= \(\frac{tan x +tan y}{1-tan x tan y}\)
3. Tan (x-y)=\(\frac{tan x -tan y}{1+tan x tan y}\)
4. Tan 2x=2 tan x/1-tan² x
5. Tan 3x= 3tan x-tan³ x/1-3 tan² x
6. Tan (90-θ )=Cot θ
7. Tan (90+θ )= -Cot θ

Example:Find the value of tan(90-45)°

Answer: We know, tan(90-θ )=cotθ

∴ tan(90-45)=cot 45°

And cot 45⁰ =1

So, tan(90-45)°=1

Example: Show that tan 3x.tan 2x.tanx= tan 3x-tan 2x-tan x

Answer: We can write, 3x=2x+x

Alos, tan 3x= tan (2x+x)

By the formula,tan (x+y)= \(\frac{tan x +tan y}{1-tan x tan y}\)

We can write,

tan 3x=tan (2x+x)= \(\frac{tan 2x +tan y}{1-tan 2x tan y}\)

tan 3x-tan 2x- tan x=tan 3x.tan 2x.tan x

Or tan 3x.tan 2x.tan x=tan 3x-tan 2x- tan x

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