Trigonometry Table: Formulas, Tricks, Examples - Embibe

# Trigonometry Table: Ratios, Tricks, and Solved Examples

Trigonometry Table: Trigonometry is a popular branch of Mathematics that deals with the study of triangles and the relationship between the length of sides and angles in a triangle. It has a wide range of applications in astronomy, architecture, aerospace, defence, etc. In this article, we have provided the trigonometry tables containing the values of all trigonometric ratios for the most commonly used angles.

The trigonometry table is a useful tool for finding the values of trigonometric ratios for standard angles such as 0°, 30°, 45°, 60°, and 90°. It comprises the values of trigonometric ratios – sine, cosine, tangent, cosecant, secant, cotangent, also known as sin, cos, tan, cosec, sec, and cot, respectively. Using the trigonometry table formula, students can compute trigonometric values for various other angles by understanding the patterns seen within trigonometric ratios and between angles.

## Introduction to Trigonometric Table

In simple words, the trigonometric table is a collection of the values of trigonometric ratios for the commonly used standard angles including 0°, 30°, 45°, 60°, and 90°. Sometimes it is also used to find the values for other angles like 180°, 270°, and 360° in the form of a table. Various patterns exist within trigonometric ratios and between their corresponding angles. Therefore, it is easy to predict the values of the trigonometric table and also use the table as a reference to calculate trigonometric values for other non-standard angles. The various trigonometric functions in Mathematics are sine function, cosine function, tan function, cot function, sec function, and cosec function.

Before beginning, let us try to recall the trigonometric formulas listed below.

1. $$\sin x=\cos \left(90^{\circ}-x\right)$$
2. $$\cos x=\sin \left(90^{\circ}-x\right)$$
3. $$\tan x=\cot \left(90^{\circ}-x\right)$$
4. $$\cot x= \tan \left(90^{\circ}-x\right)$$
5. $$\sec x=\operatorname{cosec}\left(90^{\circ}-x\right)$$
6. $$\operatorname{cosec} x=\sec \left(90^{\circ}-x\right)$$
7. $$\frac{1}{ \sin x}=\operatorname{cosec} x$$
8. $$\frac{1}{ \cos x}=\sec x$$
9. $$\frac{1}{\tan x}=\cot x$$

### Trigonometric Values

Trigonometry is the study of the relationship between the sides of a triangle (right-angled triangle) and its angles. The term trigonometric value is used to collectively define values of different ratios, such as sine, cosine, tangent, secant, cotangent, and cosecant in a trigonometric table. Every trigonometric ratio is connected to the sides of a right-angled triangle, and the trigonometric values are found using these ratios.

### Standard Angle Trigonometry Tables

The trigonometry ratio table is essentially a tabular collection of values for trigonometric functions of different conventional angles such as 0°, 30°, 45°, 60°, and 90°, as well as unusual angles such as 180°, 270°, and 360°. Because of the patterns that exist within trigonometric ratios and even between angles, it is simple to anticipate the values of the trigonometric ratios in a trigonometric table and use the table as a reference to compute trigonometric values for different other angles.

Trigonometric ratios – sine, cosine, tangent, cosecant, secant, and cotangent – are listed in the table. Sin, cos, tan, cosec, sec, and cot are the abbreviations for these ratios. The values of the trigonometric ratios of these standard angles are best remembered.

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### Steps to Create a Trigonometric Table

Students can follow the steps given below to make a sin cos tan table.

Step 1: Create a table with the angles $$0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}$$, and $$90^{\circ}$$ on the top row and all trigonometric functions $$\sin , \cos , \tan , \operatorname{cosec}, \mathrm{sec}$$, and cot in the first column.

Step 2: Determine the value of $$\sin$$.
Write the angles $$0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, 90^{\circ}$$ in ascending order and assign them values $$0,1,2,3,4$$ according to the order. So, $$0^{\circ} \rightarrow 0 ; 30^{\circ} \rightarrow 1 ; 45^{\circ} \rightarrow 2 ; 60^{\circ} \rightarrow 3 ; 90^{\circ} \rightarrow 4$$.

Then divide the values by $$4$$ and square root the entire value. $$0^{\circ} \rightarrow \sqrt{\frac{0}{4}}=0 ; 30^{\circ} \rightarrow \sqrt{\frac{1}{4}}=\frac{1}{2} ; 45^{\circ} \rightarrow \sqrt{\frac{2}{4}}=\frac{1}{\sqrt{2}} ; 60^{\circ} \rightarrow \sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2} ; 90^{\circ} \rightarrow \sqrt{\frac{4}{4}}=1$$.

This gives the values of sine for these five angles.

Now for the remaining three, use:
$$\sin \left(180^{\circ}-x\right)= \sin x \quad \sin \left(180^{\circ}+x\right)=\,- \sin x \sin \left(360^{\circ}-x\right)=\,- \sin x$$
This means, $$\sin \left(180^{\circ}-0^{\circ}\right)=\sin 0^{\circ} \quad \sin \left(180^{\circ}+90^{\circ}\right)=\,- \sin 90^{\circ} \sin \left(360^{\circ}-0^{\circ}\right)=\,- \sin 0^{\circ}$$

Step 3: Determine the value of $$\cos$$.
$$\sin \left(90^{\circ}-x\right)=\cos x$$
To find values for $$\cos x$$, use this formula.
For example, equals $$\left(90^{\circ}-45^{\circ}\right)= \sin 45^{\circ},\left(90^{\circ}-30^{\circ}\right)=\sin 60^{\circ}$$ and vice versa.
You can quickly determine the value of the $$\cos$$ function by using this method:

Step 4: Determine the value of $$\tan$$. We know that $$\sin$$ divided by $$\cos$$ equals the $$\tan$$.
$$\frac{{\sin }}{{\cos }} = \tan$$
Divide the value of $$\sin$$ at $$0^{\circ}$$ by the value of $$\cos$$ at $$0^{\circ}$$ to get the value of $$\tan$$ at $$0^{\circ}$$. Take a look at the sample below.
$$\tan 0^{\circ}=\frac{0}{1}=0$$
In the same way, the table would be as follows.

Step 5: Determine the value of $$\cot$$.
The reciprocal of $$\tan$$ is equal to the value of $$\cot$$. Divide $$1$$ by the value of $$\tan$$ at $$0^{\circ}$$ to get the value of $$\cot$$ at $$0^{\circ}$$. So, $$\cot 0^{\circ}=\frac{1}{0}=\infty$$ or Not Defined will be the value.
In the same way, a $$cot$$ table is provided below.

Step 6: Determine the value of $$\operatorname{cosec}$$.
The reciprocal of $$\sin$$ at $$0^{\circ}$$ is the value of $${\text{cosec}}$$.
$$\operatorname{cosec} 0^{\circ}=\frac{1}{0}=\infty$$ or Not Defined $$\operatorname{cosec} 0^{\circ}=\frac{1}{0}=\infty$$ or Not Defined
In the same way, a table for cosec is provided below.

Step 7: Determine the value of $$\mathrm{sec}$$.
All reciprocal values of $$\cos$$ can be used to calculate the value of $$\sec$$. The value of $$\sec$$ on $$0^{\circ}$$ is the inverse of the value of $$\cos$$ on $$0^{\circ}$$. As a result, the value will be $$\sec 0^{\circ}=\frac{1}{1}=1$$.

Similarly, the table for a sec is shown below.

Hence, the required trigonometric table for all the trigonometric ratios is as follows

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### Tricks to Remember Trigonometry Table

The trigonometry table can be useful in a variety of situations, and it is simple to remember. Remembering the trigonometric table is simple if you know the trigonometry table formula and trigonometry table trick, as trigonometry formulas are used to create the trigonometry ratios table.

Let’s learn how to recall the trigo table with just one hand! As illustrated in the image, give each finger the standard angles. We will count our fingers while filling the sine table, but we will just fill the data in reverse order for the cos table.

1st Step: To calculate the standard angle for the sine table, count the fingers on the left side.

2nd Step: Divide the number of fingers by four.

3rd Step: Take out the square root of the ratio.

Example 1: Because there are no fingers on the left side for $$\sin 0^{\circ}$$, we will use $$0$$. We obtain $$0$$ when we divide zero by four. We may determine the value of $$\sin 0^{\circ}=0$$ by taking the square root of the ratio.

Example 2: On the left-hand side, there are three fingers for $$\sin 60^{\circ}$$. We obtain $$\left(\frac{3}{4}\right)$$ when we divide $$3$$ by $$4$$. We may determine the value of $$\sin 30^{\circ}=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}$$ by taking the square root of the ratio $$\left(\frac{3}{4}\right)$$.

Similarly, we may fill the table with values for $$\sin 30^{\circ}, 45^{\circ}$$, and $$90^{\circ}$$.

PRACTICE QUESTIONS ON TRIGNOMETRIC TABLE

### Trigonometric Values Table for Unit Circle

The unit circle is a circle centred at the origin and always has a radius of $$1$$. The equation of the unit circle is $$x^{2}+y^{2}=1$$.
An ordered pair along the unit circle $$(x, y)$$ can also be known as $$( \cos \theta, \sin \theta)$$, since the $$r$$ value on the unit circle is always $$1$$. So, to find the trigonometric function values for $$45^{\circ}$$ you can look at the unit circle and easily see that $$\sin 45^{\circ}=\frac{\sqrt{2}}{2}, \cos 45^{\circ}=\frac{\sqrt{2}}{2}$$

With that information, we can easily find the values of the reciprocal functions.
$$45^{\circ}=\frac{2}{\sqrt{2}}=\frac{2 \sqrt{2}}{2}=\sqrt{2}, \sec 45^{\circ}=\sqrt{2}$$
We can also find the tangent and cotangent function values using the quotient identities $$\tan 45^{\circ}=\frac{\sin 45^{\circ}}{\cos 45^{\circ}}=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=1$$
$$\cot 45^{\circ}=1$$

### Solved Examples – Trigonometry Table

Q.1. If $$\beta=30^{\circ}$$, prove that $$3 \sin \beta-4 \sin ^{3} \beta=\sin 3 \beta$$.
Ans: L.H.S $$=3 \sin \beta-4 \beta$$
$$=3 \sin 30^{\circ}-430^{\circ}$$
$$=3\left(\frac{1}{2}\right)-4\left(\frac{1}{2}\right)^{3}$$
$$=\frac{3}{2}-4 \times \frac{1}{8}$$
$$=\frac{3}{2}-\frac{1}{2}$$
$$=1$$
$$\mathrm{RHS}= \sin 3 \beta$$
$$= \sin 3 \times 30^{\circ}$$
$$= \sin 90^{\circ}$$
$$=1$$
Therefore, $$\mathrm{LHS}=$$ RHS (Proved)

Q.2. If $$\sin (x+y)=1$$ and $$\cos (x-y)=\frac{3}{\sqrt{2}}$$, find $$x$$ and $$y$$.
Ans: $$\sin (x+y)=1$$
$$\Rightarrow \sin (x+y)=\sin 90^{\circ}$$, (since $$\sin 90^{\circ}=1$$)
$$\Rightarrow x+y=90^{\circ} \ldots \ldots \ldots \ldots \ldots \ldots . . .(i)$$
$$\cos \cos (x-y)=\frac{3}{\sqrt{2}}$$
$$\Rightarrow(x-y)=\cos \cos 30^{\circ}$$
$$\Rightarrow x-y=30^{\circ} \ldots \ldots \ldots \ldots \ldots \ldots \ldots(ii)$$
Adding, $$(i)$$ and $$(ii)$$, we get
$$x+y=90^{\circ}$$
$$x-y=30^{\circ}$$
$$2 x=120^{\circ}$$
$$x=60^{\circ}$$,(Dividing both sides by $$2$$)
Putting the value of $$x=60^{\circ}$$ in $$(i)$$ we get,
$$60^{\circ}+y=90^{\circ}$$
Subtract $$60^{\circ}$$ from both sides
$$60^{\circ}+y=90$$
$$y=30^{\circ}$$
Therefore, $$x=60^{\circ}$$ and $$y=30^{\circ}$$.

Q.3. Find the value of $$\frac{4}{3} 60^{\circ}+330^{\circ}-230^{\circ}-\frac{3}{4} 60^{\circ}$$
Answer: The given expression is $$\frac{4}{3} 60^{\circ}+330^{\circ}-230^{\circ}-\frac{3}{4} 60^{\circ}$$.
Putting the value of the angles using the trigonometric table, we have
$$=\frac{4}{3}\left(\sqrt{3}^{2}\right)+3\left(\frac{\sqrt{3}}{2}\right)^{2}-2\left(\frac{2}{\sqrt{3}}\right)^{2}-\frac{3}{4}\left(\frac{\sqrt{3}}{3}\right)^{2}$$
$$=\frac{4}{3} \times 3+3 \times \frac{3}{4}-2 \times \frac{4}{3}-\frac{3}{4} \times \frac{3}{9}$$
$$=4+\frac{9}{4}-\frac{8}{3}-\frac{1}{4}$$
$$=\frac{10}{3}=3 \frac{1}{3}$$

Q.4. Find the value of $$\tan 45^{\circ}$$.
Ans: $$\tan 45^{\circ}=\frac{\sin 45^{\circ}}{\cos 45^{\circ}}$$
Put the values of $$\sin 45^{\circ}$$ and $$\cos 45^{\circ}$$ from the trigonometric values table, we get
$$\tan 45^{\circ}=\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}$$
Hence, $$\tan 45^{\circ}=1$$

Q.5. Find the value of $$\sin \frac{\pi}{6}$$.
Ans: The value of $$\sin \frac{\pi}{6}= \sin 30^{\circ}=\frac{1}{2}$$.

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### Summary of Trigonometric Table

In this article, we have discussed the trigonometric values table in degrees and radians. We learnt the steps to create the table, tricks to remember the values in it and trigonometric values table for the unit circle, and solved some examples. Also, we solved some important questions on trigonometry table Class 10.

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### Frequently Asked Questions on Trigonometry

Here, we have enlisted some of the most important frequently asked questions related to the Trigonometric Values Table. Candidates must read these questions and answers to clear out their doubts regarding the same subject.

Q1. What is trigonometry?
Ans. Trigonometry is the branch of mathematics that deals with the relationship between the sides of a triangle (right-angled triangle) and its angles.

Q.2. How do you create a trigonometric ratio table?
Ans:
The steps for creating and remembering a trigonometric table are outlined below.
1st Step: Create a table with the angles $$0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}$$, and $$90^{\circ}$$ on the top row and all trigonometric functions $$\sin , \cos , \tan , \operatorname{cosec}, \sec$$, and cot in the first column.
2nd Step: Determine the value of $$\sin$$.
3rd Step: Reverse the order of the values of $$\sin$$ to get the values of $$\cos$$.
4th Step: The values of $$\sin$$ divided by the values of $$\cos$$ give the value of $$\tan$$.
5th Step: The reciprocal of $$\tan$$ is equal to the value of $$\cot$$.
6th Step: The reciprocal of $$\sin$$ values is the value of $${\text{cosec}}$$.
7th Step: All reciprocal values of $$\cos$$ can be used to calculate the value of $$\sec$$.

Q.3. What is the trigonometric values table?
Ans: The trigonometric table is made up of the trigonometric ratios sine, cosine, tangent, cosecant, secant, and cotangent, which are all connected. The values of standard trigonometric angles such as $$0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}$$, and $$90^{\circ}$$ are found in this table.

Q.4. How do you learn values in trigonometry?
Ans: You just need to memorise the value of sine, then the value of cosine can be determined by putting sine data in the reverse order. The value of the tangent can be determined by dividing sine by cosine. The value of secant can be determined by taking the reciprocal of cosine, and the value of cosecant can be determined by the reciprocal of sine.

Q.5. What is the ratio for sine?
Ans: Sine ratios are proportions of the length of the opposite side of the angle they represent to the hypotenuse.
$$\sin \theta=\frac{\text { Opposite side }}{\text { Hypotenuse }}$$

Q.6. What are the three basic trigonometric ratios?
Ans: There are three basic trigonometric ratios: sine, cosine, and tangent.
By using these, we can determine the values of the other three trigonometric ratios by using the relationship
$$\frac{1}{ \sin x}=\operatorname{cosec} x$$
$$\frac{1}{\cos x}= \sec x$$
$$\frac{1}{ \tan x}= \cot x$$

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Now that you know everything about the trigonometry table formula, try to create your own sin cos tan table and keep all the tips and tricks in mind. You can refer to the solved examples listed in this article while appearing for your trigonometry tests.

We hope this detailed article on the trigonometry table helps you. If you have any queries, feel free to ask in the comment section below. We will get back to you at the earliest. Till then, stay tuned to Embibe for all updates on the trigonometry table, exam preparation tips, and the latest academic articles!

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