Trigonometric Table

Trigonometric 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°. Trigonometric table comprises of all six trigonometric ratios: sine, cosine, tangent, cosecant, secant, cotangent. Trigonometric functions, also known as goniometric functions, angle functions, or circular functions, are functions that establish the relationship between an angle to the ratio of two of the sides of a right-angled triangle. Trigonometric functions are useful to study various types of angles, triangles and other objects.

Trigonometry Table Definition

The tabular arrangement for the values of all six trigonometric functions for their common angles is called a trigonometric table. The value of various trigonometric ratios can be learned using the table provided below:

Trigonometric Functions

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,

• Sine: It is defined as the ratio of perpendicular and hypotenuse and It is represented as sin θ
• Cosine: It is defined as the ratio of base and hypotenuse and it is represented as cos θ
• Tangent: It is defined as the ratio of sine and cosine of an angle. Thus the definition of tangent comes out to be the ratio of perpendicular and base and is represented as tan θ
• Cosecant: It is the reciprocal of sin θ and is represented as cosec θ.
• Secant: It is the reciprocal of cos θ and is represented as sec θ.
• Cotangent: It is the reciprocal of tan θ and is represented as cot θ.

Tricks To Learn Trigonometry 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 Create a Trigonometry 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 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 sin function divided by 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 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 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 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

Arrange all the values obtained in a table to get the trigonometric table.

Trigonometry Table Formulas

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

Identities of Complementary Angles

• sin (90° – θ) = cos θ
• cos (90° – θ) = sin θ
• tan (90° – θ) = cot θ
• cot (90° – θ) = tan θ
• sec (90° – θ) = cosec θ
• cosec (90° – θ) = sec θ

Identities of Supplementary Angles

• sin (180° – θ) = sin θ
• cos (180° – θ) = – cos θ
• tan (180° – θ) = – tan θ
• cot (180° – θ) = – cot θ
• sec (180° – θ) = – sec θ
• cosec (180° – θ) = – cosec θ

Trigonometric Identities

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,

• sin² θ + cos² θ = 1
• sec² θ – tan² θ = 1
• cosec² θ – cot² θ = 1

These three identities are the most used trigonometric identities which are widely used to solve a number of complex problems.

Solved Examples on Trigonometry

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

FAQs on Trigonometry Table

Question 1: What is Trigonometry?

Answer:

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

Question 2:  What is a trigonometric table?

Answer:

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

Question 3: What are standard angles in a trigonometric table?

Answer: 

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

Question 4: What is the value of tan 45 degrees?

Answer:

The value of tan 45 degrees is 1.

Question 5: How to learn trigonometry tables?

Answer:

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.