Trigonometry Table | Trigonometric Ratios and Formulas

Trigonometry Table is a standard table that helps us to find the values of trigonometric ratios for standard angles such as 0°, 30°, 45°, 60°, and 90°. It consists of all six trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent.

Let’s learn about the trigonometry table in detail.

Trigonometry Table

Trigonometric table is the arrangement of the values of all six trigonometric functions for their common angles in a tabulated form.

Note– Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of right-angled triangles.

Trigonometric Functions Table

Trigonometry has 6 basic trigonometric functions they are sine, cosine, tangent, cosecant, secant, and cotangent. Now let’s look into the trigonometric functions.

For, any right-angle triangle with perpendicular(P), Base(B), and Hypotenuse(H) the six trigonometric functions are as follows,

Table of Trigonometric Functions
Function	Definition	Representation	Relationship to Sides of a Right Triangle
Sine	Ratio of perpendicular and hypotenuse	sinθ	Opposite side / Hypotenuse
Cosine	Ratio of base and hypotenuse	cosθ	Adjacent side / Hypotenuse
Tangent	Ratio of sine and cosine of an angle	tanθ	Opposite side / Adjacent side
Cosecant	Reciprocal of sin θ	cscθ or cosecθ	Hypotenuse / Opposite side
Secant	Reciprocal of cos θ	secθ	Hypotenuse / Adjacent side
Cotangent	Reciprocal of tan θ	cotθ	Adjacent side / Opposite side

Note– Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right-angled triangles. It involves the study and application of sine, cosine, tangent, and other trigonometric functions to solve problems in various fields.

Check: Trigonometry: Formulas, Table, Identities and Ratios

Trick To Learn Trigonometric Ratios

Study the table discussed below to learn the trigonometric ratios in an easy-to-remember manner.

Some people have curly black hair to produce beauty
sin θ (Some) = Perpendicular(people) / hypotenuse(have)
cos θ (curly) = Base(black) / hypotenuse (hair)
tan θ (to) = Perpendicular(produce) / Base(beauty)

How to Memorize Trigonometric Table

Trigonometry Table is quite easy to remember if you know all the trigonometry formulas. There is also a trick called the one-hand trick to memorize the trigonometry table.

Step 1: In the figure above, for the sine table, count the fingers on the left side for the standard angle.

Step 2: Divide the number of fingers on the left side (calculate in the 1st step) by 4

Step 3: Find the Square root of the value calculated in Step 2.

Check: Trigonometry Formulas – List of All Trigonometric Identities and Formulas

How to Create a Trig Table

Study the following steps to create the trigonometric table for standard angles.

Step 1: Create the Table

Create a table and list all the angles such as 0°, 30°, 45°, 60°, and 90°, in the top row. Enter all trigonometric functions sin, cos, tan, cosec, sec, and cot in the first column.

Step 2: Evaluate the value for all the angles of the sin function.

For finding the values of the sin function, divide 0, 1, 2, 3, and 4 by 4 and take under the root of each value, respectively as,

For, the value of sin 0° = √(0/4) = 0
Similarly, 
sin 30° = √(1/4) = 1/2
sin 45° = √(2/4) = 1/√2
sin 60° = √(3/4) = √3/2
sin 90° = √(4/4) = 1

sin 0°	sin 30°	sin 45°	sin 60°	sin 90°
0	1/2	1/√2	√3/2	1

Step 3: Evaluate the value for all the angles of the cos function

The value of the cos function is the opposite of the value of the sin function i.e. cos 0° = sin 90°, cos 30° = sin 60° and cos 45° = sin 45°, so

cos 0°	cos 30°	cos 45°	cos 60°	cos 90°
1	√3/2	1/√2	1/2	0

Step 4: Evaluate the value for all the angles of the tan function

The value of the tan function is equal to the sin function divided by the cos function, i.e. tan x = sin x / cos x. The value of all the angles in tan function is calculated as,

tan 0°= sin 0° / cos 0° = 0/1 = 0, similarly

tan 0°	tan 30°	tan 45°	tan 60°	tan 90°
0	1/√3	1	√3	Not Defined

Step 5: Evaluate the value for all the angles of the cosec function

The value of the cosec function is equal to the reciprocal of the sin function. The value of cosec 0° is obtained by taking the reciprocal of sin 0°

cosec 0° = 1 / sin 0° = 1 / 0 = Not Defined. Similarly,

cosec 0°	cosec 30°	cosec 45°	cosec 60°	cosec 90°
Not Defined	2	√2	2/√3	1

Step 6: Evaluate the value for all the angles of the sec function

The value of the sec function is equal to the reciprocal of the cos function. The value of sec 0° is obtained by taking the reciprocal of cos 0°

sec 0° = 1 / cos 0° = 1 / 1 = 1. Similarly,

sec 0°	sec 30°	sec 45°	sec 60°	sec 90°
1	2/√3	√2	2	Not Defined

Step 7: Evaluate the value for all the angles of the cot function

The value of the cot function is equal to the reciprocal of tan function. The value of cot 0° is obtained by taking the reciprocal of tan 0°

cot 0° = 1 /tan 0° = 1 / 0 = Not defined. Similarly,

cot 0°	cot 30°	cot 45°	cot 60°	cot 90°
Not Defined	√3	1	1/√3	0

In this way, we can create the following trigonometric ratios table:

Degrees and Radians Trigonometric Table
Angle (in degrees)	Angle (in radians)	Sin	Cos	Tan	Cosec	Sec	Cot
0°	0	0	1	0	Undefined	1	Undefined
30°	π/6	1/2	√3/2	1/√3	2	2/√3	√3
45°	π/4	√2/2	√2/2	1	√2	√2	1
60°	π/3	√3/2	1/2	√3	2/√3	2	1/√3
90°	π/2	1	0	Undefined	1	Undefined	0

Trigonometric Formulas

Let’s learn about some trigonometry formulas related to Complementary and Supplementary Angles.

• Complementary Angles: Pair of angles whose sum is equal to 90°
• Supplementary Angles: Pair of angles whose sum is equal to 180°

Check: Trigonometric Ratios

Trig Identities of Complementary Angles

The identities of complementary angles are based on the relationship between the trigonometric functions of two angles that sum up to 90 degrees (or π/2 radians). These are known as co-function identities.

Trigonometric Function	Identity
Sine	sin(90°−θ)=cosθ
Cosine	cos(90°−θ)=sinθ
Tangent	tan(90°−θ)=cotθ
Cotangent	cot(90°−θ)=tanθ
Secant	sec(90°−θ)=cscθ
Cosecant	cosec(90°−θ)=secθ

Trig Identities of Supplementary Angles

The identities of supplementary angles relate to the trigonometric functions of two angles that sum up to 180 degrees (or π radians).

Trigonometric Function	Identity
Sine	sin(180°−θ)=sinθ
Cosine	cos(180°−θ)=−cosθ
Tangent	tan(180°−θ)=−tanθ
Cotangent	cot(180°−θ)=−cotθ
Secant	sec(180°−θ)=−secθ
Cosecant	cosec(180°−θ)=cosecθ

Trigonometric Identities Table

Trigonometric Identities are the identities that are highly used in solving trigonometric problems. There are various trigonometric identities but the three main trigonometric identities are,

Table of Trigonometric Identities
Trigonometric Identity	Formula
Pythagorean Identity	sin2 θ + cos2 θ = 1
Secant-Tangent Identity	sec2 θ – tan2 θ = 1
Cosecant-Cotangent Identity	cosec2θ – cot2 θ = 1

Trigonometric Table Examples

Let’s solve some questions on the trigonometric table.

Example 1: If sin θ = 4/5, then find all the trigonometric values.

Solution:

Here we have, 

sin θ = 4/5

as, sin θ = Perpendicular / Hypotenuse

so we have Perpendicular (P)= 4 and hypotenuse(H) = 5

So as per the Pythagoras theorem H² = P²+B²

Lets find out the value of base (B)

5² = B² + 4²
25 = B² + 16
25 -16 = B²
B²  = 9
B = 3

Now we have,

Sin θ = Perpendicular/Hypotenuse 
         = AB/AC = 4/5

Cosine θ = Base/Hypotenuse 
               = BC/AC = 3/5

Tangent θ = Perpendicular/Base 
                 = AB/BC = 4/3

Cosecant θ = Hypotenuse/Perpendicular 
                  = AC/AB = 5/4

Secant θ = Hypotenuse/Base 
= AC/BC = 5/3

Cotangent θ = Base/Perpendicular 
                    = BC/AB = 3/4

Example 2: Find the value of cos 45° + 2 sin 60° – tan 60°.

Solution:

From the trigonometry table, 

cos 45° = 1/√2, sin 60° = √3/2 and tan 60° = √3

Thus, 

cos 45° + 2 sin 60° – tan 60° = 1/√2 + 2(√3/2) – √3
                                              = 1/√2

Example 3: Find the value of cos 75°.

Solution:

We know that,

cos 75° = cos (45° + 30°) {as, cos (A + B) = cos A cos B – sin A sin B}
             = cos 45° cos 30° – sin 45° sin 30°
             = 1/√2 × √3/2 – 1/√2 × 1/2 
             = (√3 – 1)/2√2

cos 75°= (√3 – 1)/2√2.

Conclusion – Trigonometry Table

The Trigonometry Table provides a comprehensive reference for the trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent, along with their respective values for various angles. It serves as a valuable tool for solving trigonometric equations, analyzing geometric relationships, and understanding the behavior of periodic phenomena. Whether in mathematics, physics, engineering, or other fields, the trigonometry table aids in calculations, problem-solving, and visualization, contributing to a deeper understanding of trigonometric concepts and their applications in real-world scenarios.

Trigonometry Table – FAQs

What is Trigonometry?

Trigonometry is the branch of mathematics that deals with angle and sides of any triangle.

What is a Trigonometric Table?

Trigonometry table is a table that contains the values of all six trigonometric functions for the common angles.

Who Invented Trigonometry Table?

The Greek astronomer Hipparchus (127 BC) invented the trigonometry table.

What are Standard Angles in a Trigonometric Table?

The standard angle in a trigonometric table are 0°, 30°, 45°, 60°, and 90°

What is the value of tan 45 degrees?

The value of tan 45 degrees is 1.

How to learn Trigonometry Table?

The trick for learning trigonometric table is,

• You have to learn all the values of all the angles of sin function.
• The value of all angles of the cos function is the mirror image of the sin function.
• The values of tan function can be calculated by dividing the sin function by the cos function.
• The value of cosec function is reciprocal of sin.
• Similarly, the sec and cot are reciprocal of the cos and cot function.

What are six Basic Functions in Trigonometric Table?

The six basic trigonometric functions in the trigonometric table are Sine, Cosine, Tangent, Secant, Cotangent, and Cosecant.

Are there calculators that can replace trigonometry tables?

Scientific calculators can calculate trigonometric ratios for any angle8.

What is the use of a Trigonometry Table?

The Trigonometry Table is basically used to find the values of all trigonometric ratios for all angles. These values have a number of real life applications.