TABLE OF TRIGONOMETRY AND TRIGONOMETRIC RATIOS

TABLE OF TRIGONOMETRY

Trigonometry is a branch of mathematics that deals with the length of lines and angles of a triangle. It has a large number of applications.

The trigonometric table helps to find the standard values of trigonometric angles. The six trigonometric ratios are sin, cos, tan, cosec, sec and cot. These ratios help us to solve the problems in trigonometry. 

The trigonometric table below shows the value for standard angles   0°, 30°, 45°, 60° and 90°.

The values π/6, π/4, π/3 and π/2 are angles in radians where π = 180°.

Now, before we go any further, let us understand what ratio makes what trigonometric angle. 

Take a look at the right-angled triangle below:

This is a 90° triangle. The longest side in a right triangle is the side opposite the 90° angle. It  is called the hypotenuse.

In a right-angled triangle, it is possible to find the value of one of the sides if the value of the other two sides is given. Here, in the above diagram, the side opposite the angle Ɵ is the perpendicular and the side adjacent to the angle is the base. We can use the Pythagoras theorem for this, 

\text { perpendicular }^{2}+\text { base }^{2}=\text { hypotenuse }^{2}

a^{2}+b^{2}=c^{2}

The trigonometric ratios are the ratios of different sides of a triangle. 

For example: 

sin Ɵ = side opposite the angle Ɵ/ hypotenuse or perpendicular/ hypotenuse. 

In this case, sin Ɵ = a/c. 

Similarly, see the table below to find the value of different trigonometric ratios:

HOW TO CREATE A TRIGONOMETRIC TABLE?

It is very easy to remember the values of the ratios at the standard angles if we know the basic trigonometry formulas. 

1. sin Ɵ = cos (90° – Ɵ)
2. cos Ɵ = sin (90° – Ɵ)
3. tan Ɵ = cot (90° – Ɵ)
4. cot Ɵ = tan (90° – Ɵ)
5. cosec Ɵ = sec (90° – Ɵ)
6. sec Ɵ = cosec (90° – Ɵ)
7. 1/sin Ɵ = cosec Ɵ
8. 1/cos Ɵ = sec Ɵ
9. 1/tan Ɵ = cot Ɵ

Create a table with the standard angles in the top row and write the six trigonometric ratios in the left-most column. 

First, we will find the value of sin.

Under the angles 0°, 30°, 45°, 60° and 90°, write the numbers 0, 1, 2, 3 and 4. Divide all the numbers by four and find the root. You will get the following:

This is how we get the values of sin. 

Now to find the values of cos, we need to write the numbers in order of 4, 3, 2,1 and 0 and divide them by 4 under the root.

You will get the following values-

Note: the values of sin and cos for standard angles are in the opposite sequence of each other.

We know that tan Ɵ = sin Ɵ/cos Ɵ

We will simply divide the values of sin and cos of the standard angles to find the values of tan –

Once we know these three ratios, it is easy to find cosec, sec and cot.

i. The inverse of sin is cosec

ii. The inverse of cos is sec and

iii. The inverse of tan is cot.

We find cosec using the formula 1/sin Ɵ = cosec Ɵ

We can find sec using the formula 1/cos Ɵ = sec Ɵ

We can find cot either using 1/tan Ɵ = cot Ɵ or cot Ɵ = cosec Ɵ/ sec Ɵ