Tan 90°: Definition, Value & Real-World Applications

Imagine a rocket taking off from the ground vertically. The angle between the rocket and the ground approaches 90° as it rises. In trigonometry and scientific applications, it is essential to understand the tangent function at this angle.

In this article, we’ll explore:

• What is Tan 90°?
• What is the value of Tan 90°?
• Where is Tan 90° applied in real life?

Tangent

Tangent, written as tan, is a fundamental trigonometric function that represents the ratio of the opposite side to the adjacent side in a right-angled triangle.

Tan θ = sin θ / cos θ = Opposite / Adjacent

What is Tan 90°?

The value of the tangent trigonometric function for an angle of 90° between the opposite side and the adjacent side is known as Tangent 90 degree. The formal notation for the tangent of a ninety-degree angle is tan 90°.

Value of Tan 90°

The value of tan 90° is undefined. This is because, in a right-angled triangle, when the angle reaches 90°, the adjacent side becomes zero. Since division by zero is not defined, tan 90° is considered undefined.

Tan 90° = Opposite / 0 = Undefined

Value of Tan 90° in Radians

θ in radians = (π / 180°) × θ in degrees

90° × (π / 180°) = π / 2

Therefore, Tan 90° = Tan(π / 2) = Undefined

Value of Tan 90° using a Unit Circle

On a unit circle, the hypotenuse is always 1. We know that:

Cos θ = Adjacent / Hypotenuse

As θ approaches 90°, sin θ becomes 1 and cos θ becomes 0.

Tan 90° = Sin θ / Cos θ = 1 / 0 = Undefined

Value of Tan 90° using Trigonometric Identities

• Tan 90° = sin 90° / cos 90°
• Tan 90° = sin 90° / √(1 − sin²90°)
• Tan 90° = √(1 − cos²90°) / cos 90°
• Tan 90° = 1 / √(cosec²90° − 1)
• Tan 90° = √(sec²90° − 1)
• Tan 90° = 1 / cot 90°

Examples to Make It Easy

Example 1:

Find the value of tan 90° + cot 90°

Solution: tan 90° + cot 90° = Undefined + 0 = Undefined

Example 2:

Solve tan²90° − sec²90°

Solution: Since both tan 90° and sec 90° are undefined, the expression remains undefined.

Example 3:

A vertical flagpole casts no shadow at noon. What is the angle of elevation of the sun, and how does it relate to tan 90°?

Solution: At noon, the sun is directly overhead, making the angle of elevation 90°. The tangent function at this angle is undefined because the shadow (adjacent side) is zero.

Practice Questions

1. Using the identity 1 + tan²θ = sec²θ, explain why tan 90° is undefined.
2. Find tan 90° × cot 90°.
3. A rocket launches vertically. What is the angle between the rocket's path and the ground?

Real-Life Applications of Tan 90°

• Rocket Launches: When a rocket takes off vertically, its path makes a 90° angle with the ground.
• Architecture: Verticality in buildings relies on perpendicular angles (90°) for stability.
• Sun Elevation: At solar noon, the sun is directly overhead, forming a 90° angle with the ground.
• Surveying: Theodolites measure vertical angles, including 90°, for height and distance calculations.
• Physics of Free Fall: A free-falling object moves straight down at a 90° angle relative to the ground.

Tan 90° is a unique trigonometric function that remains undefined because it involves division by zero. While it doesn’t have a finite value, its concept is fundamental in physics, engineering, and everyday scenarios. Understanding tan 90° helps solve practical problems in motion analysis, surveying, and architecture.