Tan 90 Degrees: Value of tan 90 with Proof, Examples and FAQ

Tan 90 degrees

The value of the tangent of the angle 90°in the right-angled triangle is called tan of angle 90 degrees. 

Value of Tan 90°

The value of tan (90°) is infinity (∞) or not defined.

tan (90°) = tan π/2 = ∞ (not defined)

Proof

The exact value of tan π/2 can be derived using a unit circle and by the trigonometric approach.

• Unit circle

We can derive the value of tan (90°) by using a unit circle. The unit circle is a circle that has a radius equal to 1.

Because the radius is 1, we can easily measure the values of sin, cos and tan.

The figure shown above is a graphical representation of sin, cos and tan values in a unit circle.  When the angle is 0°, this will change the figure given below.

Now, let us see the condition when the angle is 90°

From the above figure, it is clear that when the angle is 90°, we can not define the value of tan. Therefore, the value of tan (90°) is undefined.

Alternate method: 

Since the angle is 90 degrees, there will not be any particular triangle but a unit circle. Any point at 90° can be described as (0,y) because the value at X-axes will be zero.

Unit circle has a radius of 1 which means the height (y) is 1. 

Therefore, tan (90°) = y/x

tan (90°) = 1/0

It is undefined because you can’t divide any number by zero.

Hence tan (90°) = ∞ or not defined.

• Trigonometric Method

We can prove the value of tan (90°) with a trigonometric approach.

we know that sin 90° = 1 and cos 90° = 0

Also, by trigonometric identities,

sin x/cos x = tan x

Put x = 90°

tan (90°) = sin (90°)/cos (90°)

Put the values of sin 90° and cos 90°

tan (90°) = 1/0 which can not be defined.

Some Important Formulas

tan (90° – θ ) = cot θ

tan (90° + θ ) = – cot θ

tan (-θ )= – tan θ

\operatorname{Tan} 2 x=2 \tan x /\left(1-\tan ^{2} x\right)
\operatorname{Tan} 3 x=\left(3 \tan x-\tan ^{3} x\right) /\left\{\left(1-\left(3 \tan ^{2} x\right)\right\}\right.

Example

1. Evaluate: tan (90 – 30)°

Solution:

We know that tan (90° – θ ) = cot θ

So, tan (90 – 30)° = cot 30°

and cot 30° = √3

Therefore,  tan (90 – 30)° = √3

2. Evaluate 2 tan 135° – 2 sin 30°

Solution:

tan (135°) = tan (90 + 45)°

We know that tan (90 + θ) = -cot θ

So, tan (90 + 45)° = -cot 45° = -1

and sin 30° = 1/2

Substituting the values, 

2 tan 135° – 2 sin 30°

= 2 (-1) – 2(½)

= -2 – 1

= -3