Tan Calculator

Calculate the tangent (tan) of any angle in degrees instantly with our free online Tan Calculator. Perfect for GCSE and A-Level students, engineers, surveyors, architects, builders and anyone working with trigonometry.

✔ Degrees & radians • ✔ Accurate tangent values • ✔ Instant calculation • ✔ Free calculator

What is a Tan Calculator?

A Tan Calculator calculates the tangent (tan) of any angle in seconds. Tangent is one of the three fundamental trigonometric functions alongside sine (sin) and cosine (cos), and is widely used in mathematics, engineering, construction, navigation, surveying and physics.

Simply enter an angle in degrees and the calculator instantly converts it into radians, determines the quadrant, calculates the tangent value and displays the angle on the unit circle.

What is Tangent?

In a right-angled triangle, tangent is the ratio of the opposite side to the adjacent side.

Tangent can also be calculated using sine and cosine:

For example, if the opposite side measures 5 units and the adjacent side measures 10 units:

• tan(θ) = 5 ÷ 10
• tan(θ) = 0.5

Understanding Tangent on the Unit Circle

Unlike sine and cosine, tangent is not represented by a coordinate on the unit circle. Instead, it is calculated by dividing the Y-coordinate (sine) by the X-coordinate (cosine).

• X-coordinate = cos(θ)
• Y-coordinate = sin(θ)
• tan(θ) = sin(θ) ÷ cos(θ)

As the cosine approaches zero, the tangent value becomes extremely large. This is why tan(90°) and tan(270°) are mathematically undefined.

Common Tangent Values

Angle	Radians	Tangent
0°	0	0.000000
30°	π/6	0.577350
45°	π/4	1.000000
60°	π/3	1.732051
90°	π/2	Undefined
180°	π	0.000000

Worked Examples

Example 1 — tan(45°)

• Angle = 45°
• Radians = 0.785398
• Quadrant = I
• Result = 1.000000

Example 2 — tan(30°)

• Angle = 30°
• Radians = 0.523599
• Quadrant = I
• Result = 0.577350

Example 3 — tan(135°)

• Angle = 135°
• Radians = 2.356194
• Quadrant = II
• Result = -1.000000

Where is Tangent Used?

Real-World Example: Roof Pitch

Builders frequently use tangent when calculating roof heights.

If half of a roof spans 4 metres and the roof angle is 35°, the roof rises approximately 2.8 metres.

Sine, Cosine and Tangent

Function	Definition
sin	Opposite ÷ Hypotenuse
cos	Adjacent ÷ Hypotenuse
tan	Opposite ÷ Adjacent

Things to Remember

• Tangent has no maximum or minimum value.
• tan(90°) and tan(270°) are mathematically undefined.
• Tangent is positive in Quadrants I and III.
• Tangent is negative in Quadrants II and IV.
• The calculator supports negative angles and angles larger than 360°.

Frequently Asked Questions

What is tan(45°)?

tan(45°) = 1

Why is tan(90°) undefined?

Because cosine equals zero at 90°, and dividing by zero is mathematically impossible.

Can tangent be negative?

Yes. Tangent is negative in Quadrants II and IV.

Does this calculator use degrees?

Yes. Enter the angle in degrees and the calculator performs all conversions automatically.

Can tangent become infinitely large?

As an angle approaches 90° or 270°, the tangent value increases or decreases without bound, which is why these angles are considered undefined.

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