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Trigonometric Table

The trigonometric ratios table helps in finding the values of standard angles such as 0°, 30°, 45°, 60°, and 90°. Trigonometric table comprises trigonometric ratios – sine, cosine, tangent, cosecant, secant, cotangent. These ratios, in short, are written as sin, cos, tan, cosec, sec, and cot. The values of trigonometric ratios of standard angles are essential in solving trigonometry problems.

Trigonometry deals with the measurement of angles and problems related to angles. The term trigonometry is derived from a Latin word. Trigonometry makes use of the relationship between the angles and sides of a triangle.

1.	What Is Trigonometric Table?
2.	Trigonometric Values
3.	Trigonometry Table for Standard Angles
4.	Learn Trigonometric Table
5.	Trigonometry Table in Radians
6.	Solved Examples
7.	Practice Questions
8.	FAQs on Trigonometric Table

What Is Trigonometric Table?

The Trigonometric Table is simply a collection of trigonometric values of various standard angles including 0°, 30°, 45°, 60°, 90°, sometimes with other angles like 180°, 270°, and 360° included, in a tabular format. Because of patterns existing within trigonometric ratios and even between angles, it is easy to both predict the values of the table and use the table as a reference to calculate trigonometric values for various other angles.
The trigonometric functions are namely sine function, cosine function, tan function, cot function, sec function, and cosec function.

Trigonometric Values

Trigonometry deals with the relationship between the sides of a triangle (right-angled triangle) and its angles. The trigonometric value is a collective term for values of different ratios, such as sine, cosine, tangent, secant, cotangent, and cosecant. All the trigonometric functions are related to the sides of a right-angle triangle and their values are found using the following ratios.

Trigonometric ratios

To remember this easily remember the word "SOHCAHTOA"!

SOH

Sine = Opposite / Hypotenuse

CAH

Cosine = Adjacent / Hypotenuse

TOA

Tangent = Opposite / Adjacent

Standard Angles

The trigonometric values for the angles 0°, 30°, 45°, 60°, and 90° are commonly used to solve the trigonometry problems. These values are associated with the measurement of the lengths and the angles of a right-angle triangle. Thus, 0°, 30°, 45°, 60°, and 90° are called the standard angles in trigonometry.

Trigonometry Table for Standard Angles

The trigonometric table is simply a collection of values of trigonometric functions of various standard angles including 0°, 30°, 45°, 60°, 90°, along with with other angles like 180°, 270°, and 360° included, in a tabular format. It is easy to predict the values of the table and to use the table as a reference to calculate trigonometric values for various other angles, due to the patterns existing within trigonometric ratios and even between angles,

The table consists of trigonometric ratios – sine, cosine, tangent, cosecant, secant, and cotangent. In short, these ratios are written as sin, cos, tan, cosec, sec, and cot. It is best to remember the values of the trigonometric ratios of these standard angles.

trigonometric table

Using the above table, values of sin π/6, cos π/6, tan π/6, sin π/4, cos π/4, tan π/4, sin π/3, cos π/3, tan π/3, sin π/2, cos π/2, and tan π/2 in radians could be easily found out. A trigonometric table has wide application in fields like science and engineering.

A few key points that can be noted in the trigonometric table are,

The values of complementary angles present like 30° and 60° can be computed using complementary formulas for the various trigonometric ratios.
The value for some ratios given as ∞. The reason for this is that while computing the values, a "0" appears in their denominator, so the value becomes undefined and is said to be equivalent to infinity.
There is a sign change in the values in various places under 90°, 180°, etc. This is due to the change in different quadrants.

Learn Trigonometric Table

The trigonometric table might seem complex at first, but it can be learned easily by only the values of sine for the 8 standard angles. Before generating the table, there are few formulas that must be followed as given below,

tan x = sin x/cos x
cosec x = 1/sin x
sec x = 1/cos x
cot x = 1/tan x

Given below are the steps to create and remember a trigonometric table.

Step 1: Create a table with the top row listing the angles such as 0°, 30°, 45°, 60°, 90°, and write all trigonometric functions in the first column such as sin, cos, tan, cosec, sec, cot.
Step 2: Determining the value of sin. Write the angles 0°, 30°, 45°, 60°, 90° in ascending order and assign them values 0, 1, 2, 3, 4 according to the order. So, 0° ⟶ 0; 30° ⟶ 1; 45° ⟶ 2; 60° ⟶ 3; 90° ⟶ 4. Then divide the values by 4 and square root the entire value. 0° ⟶ √0/2; 30° ⟶ 1 /2; 45° ⟶ 1/ √2; 60° ⟶ √3/2; 90° ⟶ √(4/4). This gives the values of sine for these 5 angles. Now for the remaining three use:

sin (180° − x) = sin x
sin (180° + x) = -sin x
sin (360° − θ) = -sin x

This means,
sin (180° − 0) = sin 0º
sin (180° + 90) = -sin 90º
sin (360° − 0) = -sin 0º

Angles(in Degrees)	0°	30°	45°	60°	90°	180°	270°	360°
sin	0	1/2	1/√2	√3/2	1	0	-1	0

Step 3: Determining the value of cos. sin (90° – x) = cos x. Use this formula to compute values for cos x. For example, cos 45° = sin (90° – 45°) = sin 45°. Similarly, cos 30° = sin (90°-30°) = sin 60°. Using this, you can easily find out the value of cos function as,

Angles(in Degrees)	0°	30°	45°	60°	90°	180°	270°	360°
cos	1	√3/2	1/√2	1/2	0	-1	0	1

Step 4: Determining the value of tan. (tan x = sin x/cos x). Hence, the value of tan function can be generated as,

Angles(in Degrees)	0°	30°	45°	60°	90°	180°	270°	360°
tan	0	1/√3	1	√3	∞	0	∞	0

Step 5: Determining the value of cot. (cot x = 1/tan x). Use the relation to generate the cot function as,

Angles(in Degrees)	0°	30°	45°	60°	90°	180°	270°	360°
cot	∞	√3	1	1/√3	0	∞	0	∞

Step 6: Determining the value of cosec. (cosec x = 1/sin x)

Angles(in Degrees)	0°	30°	45°	60°	90°	180°	270°	360°
cosec	∞	2	√2	2/√3	1	∞	-1	∞

Step 7: Determining the value of sec. (sec x = 1/cos x)

Angles(in Degrees)	0°	30°	45°	60°	90°	180°	270°	360°
sec	1	2/√3	√2	2	∞	-1	∞	-1

The following table covers the value of trigonometric ratios for all basic angles ranging from 0º to 360º.

trigonometric ratios table

Trick to Remember Trigonometric Table

Let us learn the one-hand trick for remembering the trigonometric table easily! Designate each finger the standard angles as shown in the image. For filling the sine table we will include counting of the fingers, while for the cos table we will simply fill the values in reverse order.

Trick to remember trigonometric table

Step 1: For the sine table, count the fingers on the left side for the standard angle.
Step 2: Divide the number of fingers by 4
Step 3: Take out the square root of the ratio.

Example 1: For sin 0°, there are no fingers on the left-hand side, therefore we will take 0. Dividing the zero by 4 we get 0. Taking the square root of the ratio we would get the value of sin 0° = 0

Example 2: For sin 60°, there are 3 fingers on the left-hand side. Dividing 3 by 4 we get (3/4). Taking the square root of the ratio √(3/4) we would get the value of sin 30° = √3/2.

Similarly, we can find out values for sin 30°, 45°, and 90° to fill the table.

trigonometric table

In order to complete the cos table, simply fill in the values of sine in a reverse manner. For tan values, create the ratios by taking numerators of sine and cos values of the corresponding angle as indicated by the arrows in the above picture.

Given below is a list of trigonometric formulas that would help you memorize the trigonometric table, based on the relationship between different trigonometric ratios.

sin x = cos (90° – x)
cos x = sin (90° – x)
tan x = cot (90° – x)
cot x = tan (90° – x)
sec x = cosec (90° – x)
cosec x = sec (90° – x)
1/sin x = cosec x
1/cos x = sec x
1/tan x = cot x

Trigonometry Table in Radians

The table below presents the values of trigonometric ratios in radians.

Trigonometric table

Important Notes

Trigonometric values are based on the three major trigonometric ratios: Sine, Cosine, and Tangent.
Sine or sin θ = Side opposite to θ / Hypotenuse
Cosines or cos θ = Adjacent side to θ / Hypotenuse
Tangent or tan θ =Side opposite to θ / Adjacent side to θ
0°, 30°, 45°, 60°, and 90° are called the standard angles in trigonometry.
The angle values of trigonometric functions cotangent, secant, cosecant, can also be calculated using these standard angles values of sine, cosecant, tangent.
All the higher angle values of trigonometric functions such as 120°, 390°, .... can be easily calculated from the standard angle values.

Example 1: Find out the exact value of sin15º.

Solution:

Using the trigonometric table, we know that sin45º = 1/√2, cos30º = (√3/2), cos45º = 1/√2, and sin30º = 1/2

sin15º = sin(45º - 30º) = sin45ºcos30º - cos45ºsin30º = (√2/2) • (√3/2) + (√2/2) • (1/2) = (√6 + √2)/4 = (√3 + 1)/2√2

Answer: The value of sin15º = (√3 + 1)/2√2
Example 2: Can you determine the height of the glacier given below? Use of calculators is permitted.

Solution:

The length of the hypotenuse is known to be 20 units. We want to find out the opposite side to the angle. Sine ratio uses the opposite side and hypotenuse. Remember the word "SOHCAHTOA"! SOH: Sine = Opposite / Hypotenuse. Thus, sin45º = Opposite/Hypotenuse, which gives 1/√2 = Opposite/20. Hence, Opposite = 20/√2, Opposite ≈ 14.14. ∴ The height of the glacier is approx. 14.14 units.

Answer: Hence, the height of the glacier is approx. 14.14 units.
Example 3: Use the trigonometric table to write the values of:

(a) sin(π/6)
(b) sec(π/3)
(c) cot(π/2)

Solution:

The trigonometric ratios table helps us to find the above values quickly. We have:
(a) sin(π/6) = sin30º = 1/2
(b) sec(π/3) = sec60º = 2
(c) cot(π/2) = cot90º = 0

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FAQs on Trigonometric Table

What Is the Trigonometric Table?

Trigonometric ratios table or sin cos table gives the values of trigonometric functions for the standard angles such as 0°, 30°, 45°, 60°, and 90°, in a tabulated manner.

What Are the Standard Angles in a Trigonometric Table?

The angles 0°, 30°, 45°, 60°, and 90° are called standard angles, and the trigonometric values for these standard angles are commonly used to solve the trigonometry problems. These values are associated with the measurement of lengths and angles of the right-angle triangle.

What Does the Infinity Value of the Trigonometric Ratio Mean?

Undefined values are often given an infinity \((\infty )\) value because the value is such large that there is no definite value to assign them at this point.

What Is Trigonometry?

Trigonometry is the branch of mathematics that deals with the study of the relationship between the sides of a triangle (Right-angled triangle) and its angles.

What Are Trigonometric Functions and their Types?

Trigonometric functions are defined as the functions of an angle of a right-angled triangle. There are 6 basic types of trigonometric functions or ratios which are:

Sin function
Cos function
Tan function
Cot function
Cosec function
Sec function

How to Find the Value of Trigonometric Functions?

All the trigonometric functions are related to the ratios of sides of the triangle and their values can be easily found by using the following relations:

sinθ = Opposite/Hypotenuse
cosθ = Adjacent/Hypotenuse
tanθ = Opposite/Adjacent
cotθ = 1/tanθ = Adjacent/Opposite
cosecθ = 1/sinθ = Hypotenuse/Opposite
secθ = 1/cosθ = Hypotenuse/Adjacent

What Are the Six Basic Trigonometric Functions?

The six basic trigonometric functions are Sine, Cosine, Tan, Sec, Cot, and Cosec function.

How Does a Trigonometry Table Work?

The trigonometric ratios table helps us to find the values of trigonometric standard angles such as 0°, 30°, 45°, 60°, and 90°. It comprises trigonometric ratios of sine, cosine, tangent, cosecant, secant, cotangent. These ratios are written in short as sin, cos, tan, cosec, sec, and cot.

Who Invented Trigonometry Table?

The Greek astronomer Hipparchus (127 BC) is believed to be the first to make a table of trigonometric functions (based on the chords in a circle).

What Is the Easiest Way to Learn Trigonometry Tables?

Step 1: For the sine table, count the fingers on the left side for the standard angle.
Step 2: Divide the number of fingers by 4
Step 3: Take out the square root of the ratio.