Tricks to Remember Trigonometric Tables: Hand Trick, Mnemonics and Quadrant Rules

Learning the values of the trigonometric table through simple patterns and memory tricks rather than rote learning makes it easy to understand, memorize and use the values with accuracy. If you are preparing for school exams or competitive exams, these tips to remember trigonometric table values will help you to remember the values of sine, cosine, tangent, cosecant, secant and cotangent of standard angles like 0°, 30°, 45°, 60° and 90° quickly. In this guide, you will learn easy mnemonics, shortcuts and practical tips to remember the trigonometric table accurately and with confidence. 

Table of Contents

• Trick to Remember the Trigonometric Table
• Mnemonics to Memorize Trigonometric Ratios
• How to Remember Positive Trigonometric Ratios in Each Quadrant
• Frequently Asked Questions of Tricks to Remember Trigonometric Table

Trick to Remember the Trigonometric Table

This is one of the most reliable tricks, as you can turn your own hand into a calculator.

Hold up your left hand with all five fingers spread out. Assign the five standard angles to your fingers in order, starting from the thumb:

• Thumb = 90 degrees

• Index finger = 60 degrees

• Middle finger = 45 degrees

• Ring finger = 30 degrees

• Little finger = 0 degrees

Rule for Sine

Fold down the finger at the angle you want. Count how many fingers are standing below the folded finger (towards the little finger). Then:

sin(θ) = square root of fingers below, divided by 2

Rule for Cosine

Count how many fingers are standing above the folded finger (towards the thumb). Then:

cos(θ) = square root of fingers above divided by 2

Rule for Tangent

Once you have sine and cosine, tangent is simply their ratio:

tan(angle) = square root of fingers below/ square root of fingers above

= sin(θ) divided by cos(θ)

Example 1: Finding sin 60° and cos 60°

Fold your index finger, as it represents 60°.

Below the folded finger are the middle, ring and little fingers, so sin 60° = √3 / 2

Above the folded finger is only the thumb, so cos 60° = √1 / 2 = 1/2

Therefore,

tan 60° = (√3/2) ÷ (1/2) = √3

Example 2: Finding sin 0° and cos 0°

Fold your little finger, as it represents 0°.

There are no fingers below the folded finger, so sin 0° = √0 / 2 = 0

Above the folded finger are the thumb, index, middle, and ring fingers, so cos 0° = √4 / 2 = 2/2 = 1

Therefore,

tan 0° = 0 ÷ 1 = 0

The Root Over 2 Pattern

Instead of memorising five separate sine values, notice that every sine value for the standard angles follows one simple pattern:

sin(0, 30, 45, 60, 90) = √(0)/2, √(1)/2, √(2)/2, √(3)/2, √(4)/2

This simplifies to 0, 1/2, 1/√2, √3/2 and 1. For cosine, the same numbers appear, just written in reverse order, since cos(angle) = sin(90 minus angle).

Mnemonics to Memorize Trigonometric Ratios

The easiest way to remember the meanings of sine, cosine, and tangent in a right triangle is by using two popular mnemonics.

SOH CAH TOA

SOH → Sine = Opposite / Hypotenuse

CAH → Cosine = Adjacent / Hypotenuse

TOA → Tangent = Opposite / Adjacent

Each group of letters tells you which sides to divide for a particular ratio. 

Pandit Badri Prasad Har Har Bole

This sentence is widely taught in Indian schools because it covers all six ratios in one go, using Perpendicular, Base and Hypotenuse instead of opposite and adjacent:

• Pandit = sin = Perpendicular / Hypotenuse

• Badri Prasad = cos = Base / Hypotenuse

• Har Har Bole = tan = Perpendicular / Base

P from Pandit is for Perpendicular and H from Har is for Hypotenuse, B from Badri is for Base and H from Har (in the middle) is for Hypotenuse and P from Prasad is for Perpendicular and B from Bole is for Base.

How to Remember Positive Trigonometric Ratios in Each Quadrant

In each quadrant of the coordinate plane, only certain trigonometric ratios stay positive. This is called the ASTC rule and it is read starting from the first quadrant and moving anticlockwise.

The word ASTC itself, All, Sin, Tan, Cos, is the shortest mnemonic there is, but many students prefer a full sentence. Here are several that are commonly used, so pick whichever one you remember best:

• All Students Take Chocolate

• Add Sugar To Coffee

• All Silver Tea Cups

• All Students To College

• After School To College

Read more: Trigonometry Exercises with Answers for Practice