Trigonometric Table

The trigonometric ratios table helps in finding the values of trigonometric standard angles such as 0°, 30°, 45°, 60°, and 90°. It 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. Therefore, it is mandatory to remember the values of the trigonometric ratios of these standard angles.

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 are the Trigonometric Values?
2.	What is a Trigonometric Table?
3.	How to Learn Trigonometric Table?
4.	Trigonometric Table Calculator

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.

To remember this easily remember the word "SOHCAHTOA"! 

• SOH

Sine = Opposite / Hypotenuse

• CAH

Cosine = Adjacent / Hypotenuse

• TOA

Tangent = Opposite / Adjacent

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.

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.

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 trigonometry 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. 

How to 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 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.

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.

• 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.

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

Let us try to calculate the values of trigonometric functions for different angles given in degrees or radians and compare the values with the ones in the 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.

Challenging Questions

• What will be the trigonometric values of negative angles?
• A right-angled triangle has an area of 400 square units. What will be the length of its hypotenuse? 

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What is the Sin Cos 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?

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 lengths and angles of the right-angle triangle. Thus, 0°, 30°, 45°, 60°, and 90° are called the standard angles in trigonometry.

What does the Infinity Value of the Trigonometric Ratio Mean?

Undefined values are often given an infinity value because 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:

• sin = Opposite/Hypotenuse
• cos = Adjacent/Hypotenuse
• tan = Opposite/Adjacent
• cot = 1/tan = Adjacent/Opposite
• cosec = 1/sin = Hypotenuse/Opposite
• sec = 1/cos = Hypotenuse/Adjacent