Trigonometry Table: Trigonometry Formula, Examples, Tips

Basic Trigonometry Formulas and Trigonometry Table

We generally consider only right-angled triangles for trigonometry formulas but it can also be used for general triangles. 

In a right-angled triangle, we have three sides, namely – Adjacent side, Hypotenuse, and Opposite side. These three sides are shown in the right-angled triangle given below.

AB = Adjacent side to angle A

BC = Opposite side to angle A

CA = Hypotenuse to angle A

In a right triangle – 

• The longest side is the hypotenuse, 
• The side opposite to the angle is perpendicular, 
• The side where both the hypotenuse and opposite side rests is the base. 

Before moving to the trigonometric table, let us learn about the basic trigonometry formulas.

These formulas help us find the relationship between trigonometric ratios and the ratio of the corresponding sides of a right-angled triangle. 

There are six trigonometric ratios or trigonometric functions which are –

• sine (sin) 
• cosine (cos) 
• secant (sec) 
• cosecant (cosec) 
• tangent (tan) 
• cotangent (cot) 

All the trigonometric functions relate to the sides of a right-angle triangle, and we can find their formulas using the following ratios.

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

There is a shortcut trick to remember these formulas.

“Some People Have Curly Brown Hair Turned Permanently Black”

This phrase is divided into three parts. Every alphabet of the first word represents a trigonometric identity and the next two words’ first alphabet describes the formula for it.  

Like ‘S’ alphabet in ‘Some’ indicates ‘sin’ function. Now next two words’ first alphabet describes the formula for sin. ‘P’ alphabet in ‘People’ represents ‘Perpendicular’ and ‘H’ alphabet in ‘Have’ represents ‘Hypotenuse’. 

Thus we can memorize, sin (some) = Perpendicular (people) / Hypotenuse (have)

Similarly, 

cos(curly) = base (brown) / hypotenuse (hair)

tan( turned)= perpendicular(permanently) / base (black)

What is a Trigonometry Table?

The trigonometry table is a tabular representation of values of trigonometric functions of various standard angles, including 0°, 30°, 45°, 60°, 90°, 180°, 270°, and 360°.  The values of trigonometric functions of these angles are essential to solve the trigonometry problems. 

Tips to Remember Trigonometry Table

The trigonometric table may seem complex to remember but it can be remembered easily by using a trick. Before revealing the trick, there are some formulas given below that are very important to learn.

• tan x = (sin x/cos x)
• cosec x = (1/sin x)
• sec x = (1/cos x)
• cot x = (1/tan x)
• 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)

Now we will use the trick to create and remember the trigonometric table.

• Create a table and list the top row with angles such as 0°, 30°, 45°, 60°, 90°, and write the trigonometric ratio in the first column such as sin (for example).

• Now we will determine the values for sin. Write numbers 0, 1, 2, 3, 4 under the angles 0°, 30°, 45°, 60°, 90° respectively. 

• Now divide the numbers by 4 and find the square root. We will get √(0/4), √(¼), √(2/4), √(¾), and √(4/4). 

• On simplifying this, we will get the values of sine for these 5 angles. 

• Now for the remaining three angles, use the following formulas:

sin (180° − x) = sin x

sin (180° + x) = -sin x

sin (360° − x) = -sin x

This means,

sin 180° = sin (180° − 0°) = sin 0° = 0

sin 270° = sin (180° + 90°) = -sin 90° = -1

Sin 360° = sin (360° − 0°) = -sin 0° = 0

• Now we will determine the values for cos using the formula cos x = sin (90° – x). 

For example, cos 60° = sin (90° – 30°) = sin 30° = ½. 

Similarly, you can find out the other values.

• To determine the values for tan, we use the formula tan x = (sin x/cos x). 

For example, the value of tan 30° = (sin 30°/cos 30°) = (½) /(√3/2) = 1/√3. 

Similarly, we can generate the other values. 

• We can determine the values for cot using the formula cot x = (1/tan x). 

For example, the value of cot 30° = 1/tan 30° = 1/(1/√3) = √3.

• Similarly, for cosec x, we use cosec x = (1/sin x)

• For sec x, we use sec x = (1/cos x)

The value of trigonometric functions for angles ranging from 0° to 360° is given in the following trigonometry table. 

Examples

1. In the figure given below, find the value of tan A?

Solution:

tan θ = Opposite Side/Adjacent Side

So, tan A = 3/4

2. What is the value of cos 270°?

Solution:

We know that, cos x = sin (90° – x)

So, cos 270° = sin (90° – 270°) = – sin 180°

and sin 180° = sin (180° − 0°) = sin 0° = 0

Hence, cos 270° = – sin 180° = 0