Trigonometric Table (Sine, Cosine, Tangent)

This trigonometric table provides reference values for the three main trigonometric functions: sine (sin), cosine (cos), and tangent (tan). It is useful in geometry, trigonometry, physics, engineering, and many other applications where angles and triangle relationships appear.

On this page you will find a quick explanation of the functions, a reference table for common angles, special exact values, and core identities that are frequently used in calculations and homework.

What Is a Trigonometric Table?

A trigonometric table lists values of trigonometric functions for different angles. Before electronic calculators were common, tables like these were the main tool for solving problems in surveying, navigation, astronomy, and engineering. Even today, they are still used for quick checks, mental estimation, and educational purposes.

Trigonometric Functions in a Right Triangle

For an acute angle θ in a right triangle, with sides defined as:

• Opposite: side opposite the angle θ
• Adjacent: side next to θ (but not the hypotenuse)
• Hypotenuse: longest side, opposite the right angle

The basic trigonometric functions are:

• sin(θ) = opposite / hypotenuse
• cos(θ) = adjacent / hypotenuse
• tan(θ) = opposite / adjacent

Trigonometric Functions on the Unit Circle

Trigonometric functions can also be defined using the unit circle (a circle of radius 1 centered at the origin). For a point on the unit circle corresponding to an angle θ (measured from the positive x-axis):

• cos(θ) is the x-coordinate of the point.
• sin(θ) is the y-coordinate of the point.
• tan(θ) = sin(θ) / cos(θ), whenever cos(θ) ≠ 0.

Trigonometric Table (0° to 90°)

The table below shows approximate values of sine, cosine, and tangent for selected angles between 0° and 90°. Values are rounded to four decimal places.

Key Angles from 0° to 360°

The following table lists approximate values for sine, cosine, and tangent at some of the most commonly used angles around the full circle.

Special Angles with Exact Values

Some angles have exact trigonometric values that are used very frequently in trigonometry and calculus.

Core Trigonometric Identities

These identities are often used together with the values in the table.

Pythagorean Identity

sin²(θ) + cos²(θ) = 1

Tangent Identity

tan(θ) = sin(θ) / cos(θ)

Cofunction Identities

sin(90° - θ) = cos(θ)
cos(90° - θ) = sin(θ)
tan(90° - θ) = cot(θ)

Angle Sum and Difference

sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b)
cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b)

How to Use This Table

You can use this trigonometric table to solve problems involving right triangles, circles, waves, oscillations, and many other phenomena where angles are involved. For example:

• Finding heights and distances in surveying problems
• Checking homework in trigonometry and pre-calculus
• Verifying calculator results during exams or practice
• Estimating values quickly without a calculator

Related Math Tables

If you are working with exponents, logarithms, and trigonometric functions, these tables may also be useful:

• Logarithm Table (Base 10)
• Antilogarithm Table (Base 10)

Frequently Asked Questions

What are trigonometric tables used for?

Trigonometric tables are used to look up values of sine, cosine, and tangent for different angles. They are helpful in geometry, trigonometry, physics, engineering, navigation, and many other fields.

Do I still need trig tables if I have a calculator?

Yes. While calculators and computer algebra systems are very convenient, tables are still useful for quick reference, estimation, and understanding how the functions behave as the angle changes.

Why is tan(90°) undefined?

The tangent function is defined as tan(θ) = sin(θ) / cos(θ). At 90°, cos(90°) = 0, and division by zero is undefined. For this reason, tan(90°) does not have a finite value.

Are the values in this table exact?

The decimal values are approximations rounded to four decimal places. However, the special angles table lists exact values using square roots and simple fractions.