**Trigonometry Formulas:** Trigonometry is the branch of mathematics that deals with the relationship between the sides and angles of a triangle. There are many interesting applications of Trigonometry that one can try out in their day-to-day lives. For example, if you are on the terrace of a tall building of **known height** and you see a **post box** on the other side of the road, you can easily calculate the width of the road using trigonometry formulas.

Of course, you need to have an understanding of the various relationships between the sides of the triangle formed by joining the three points – you, the foot of the building, and the post box, and the angles between the sides of the triangle thus formed. You need to know the various trigonometry formulas and what they mean.

Trigonometry has immense applications in construction, flight engineering, criminology, marine biology, engineering, and tons of other branches. Students are usually introduced to the basics of Trigonometry in high school (Class 9 or Class 10). Then, they are moved into the more complex concepts covered in Class 11 and Class 12. To ensure you don’t get confused with its elements, we will provide you with the complete list of Trigonometry Formulas for Class 10, Class 11, and Class 12.

**KNOW EVERYTHING ABOUT TRIGONOMETRIC RATIOS HERE**

## Trigonometry Formulas For Class 10, 11 & 12

Let us consider the following right-angled triangle:As you can see, the three sides of the triangle are:

a. **Base:** The side that is horizontal to the plane.

b. **Perpendicular:** The side making an angle of 90 degree with the Base.

c. **Hypotenuse:** The longest side of the triangle.

Also, \(\theta\) is the angle made by Hypotenuse and Base.

Then,sine of angle \(\theta\) = \(\sin \theta\) = \(\frac{Perpendicular}{Hypotenuse}\)

cosine of angle \(\theta\) = \(\cos \theta\) = \(\frac{Base}{Hypotenuse}\)

tangent of angle \(\theta\) = \(\tan \theta\) = \(\frac{Perpendicular}{Base}\)

cotangent of angle \(\theta\) = \(\cot \theta\) = \(\frac{Base}{Perpendicular}\)

cosecant of angle \(\theta\) = \(cosec \theta\) = \(\frac{Hypotenuse}{Perpendicular}\)

secant of angle \(\theta\) = \(\sec \theta\) = \(\frac{Hypotenuse}{Base}\)

Note that, sine, cosine, tangent, cotangent, cosecant, and secant are called Trigonometric Functions that defines the relationship between the sides and angles of the triangle.

### Reciprocal Relationship Between Trigonometric Functions

The reciprocal relationship between different Trigonometric Functions are as under: \(\tan \theta\) = \(\frac{1}{\cot \theta}\) = \(\frac{\sin \theta}{\cos \theta}\) \(\cot \theta\) = \(\frac{1}{\tan \theta}\) = \(\frac{\cos \theta}{\sin \theta}\) \(cosec \theta\) = \(\frac{1}{\sin \theta}\) \(\sec \theta\) = \(\frac{1}{\cos \theta}\) |

### Trigonometric Ratios Of Complementary Angles

**First Quadrant**

sin(π/2−\(\theta\)) = \(\cos \theta\) cos(π/2−\(\theta\)) = \(\sin \theta\) tan(π/2−\(\theta\)) = \(\cot \theta\) cot(π/2−\(\theta\)) = \(\tan \theta\) sec(π/2−\(\theta\)) = cosec\(\theta\) cosec(π/2−\(\theta\)) = \(\sec \theta\) |

**Second Quadrant**

sin(π−\(\theta\)) = \(\sin \theta\) cos(π−\(\theta\)) = -\(\cos \theta\) tan(π−\(\theta\)) = -\(\tan \theta\) cot(π−\(\theta\)) = -\(\cot \theta\) sec(π−\(\theta\)) = -sec\(\theta\) cosec(π−\(\theta\)) = cosec\(\theta\) |

**Third Quadrant**

sin(π+\(\theta\)) = -\(\sin \theta\) cos(π+\(\theta\)) = -\(\cos \theta\) tan(π+\(\theta\)) = \(\tan \theta\) cot(π+\(\theta\)) = \(\cot \theta\) sec(π+\(\theta\)) = -sec\(\theta\) cosec(π+\(\theta\)) = -cosec\(\theta\) |

**Fourth Quadrant**

sin(2π−\(\theta\)) = -\(\sin \theta\) cos(2π−\(\theta\)) = \(\cos \theta\) tan(2π−\(\theta\)) = -\(\tan \theta\) cot(2π−\(\theta\)) = -\(\cot \theta\) sec(2π−\(\theta\)) = sec\(\theta\) cosec(2π−\(\theta\)) = -cosec\(\theta\) |

### Periodicity Identities

sin(2nπ + \(\theta\)) = \(\sin \theta\) cos(2nπ + \(\theta\)) = \(\cos \theta\) tan(2nπ + \(\theta\)) = \(\tan \theta\) cot(2nπ + \(\theta\)) = \(\cot \theta\) sec(2nπ + \(\theta\)) = \(\sec \theta\) cosec(2nπ + \(\theta\)) = cosec\(\theta\) |

### Trigonometry Table

**Trigonometry table** is a table that you can refer to if you are not so sure about the values of different angles. Below is the table for trigonometry formulas of different angles which are commonly used for solving various problems.

Angles (In Degrees) | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |

Angles (In Radians) | 0° | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |

sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | 0 | -1 | 0 |

cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1 | 0 | 1 |

tan | 0 | 1/√3 | 1 | √3 | ∞ | 0 | ∞ | 0 |

cot | ∞ | √3 | 1 | 1/√3 | 0 | ∞ | 0 | ∞ |

cosec | ∞ | 2 | √2 | 2/√3 | 1 | ∞ | -1 | ∞ |

sec | 1 | 2/√3 | √2 | 2 | ∞ | -1 | ∞ | 1 |

**CHECK THE COMPLETE TRIGONOMETRY TABLE HERE**

### Trigonometric Identities

\(\sin^{2}\theta + \cos^{2}\theta = 1\) \(\tan^{2}\theta + 1 = \sec^{2}\theta\) \(\cot^{2}\theta + 1 = cosec^{2}\theta\) |

### Sign Of Trigonometric Functions

\(\sin (-\theta ) = -\sin \theta\) \(\cos (-\theta ) = \cos \theta\) \(\tan (-\theta ) = -\tan \theta\) \( cosec (-\theta ) = -cosec \theta\) \(\sec (-\theta ) = \sec \theta\) \(\cot (-\theta ) = -\cot \theta\) |

### Trigonometric Functions Of Sum And Difference Of Two Angles

\(\sin (A+B) = \sin A \cos B + \cos A \sin B\) \(\sin (A -B) = \sin A \cos B – \cos A \sin B\) \(\cos (A + B) = \cos A \cos B – \sin A \sin B\) \(\cos (A – B) = \cos A \cos B + \sin A \sin B\) \(\tan (A + B) = \frac{\tan A + \tan B}{1 – \tan A \tan B}\) \(\tan (A – B) = \frac{\tan A – \tan B}{1 + \tan A \tan B}\) |

### Trigonometry Formulas Involving Product Identities

\(\sin \, A \,\ sin \, B = \frac{1}{2}\left [ \cos\left ( A – B \right ) -\cos \left ( A+B \right ) \right ]\) \(\cos\, A \, \cos\, B = \frac{1}{2}\left [ \cos \left ( A – B \right ) + \cos \left ( A+B \right ) \right ]\) \(\sin\, A \, \cos\, B = \frac{1}{2}\left [ \sin\left ( A + B \right ) + \sin \left ( A-B \right ) \right ]\) \( \cos\, A \, \sin\, B = \frac{1}{2}\left [ \sin\left ( A + B \right ) – \sin\left ( A-B \right ) \right ]\) |

### Trigonometry Formulas Involving Sum To Product Identities

\(\sin\, A + \sin \, B = 2\, \sin \left ( \frac{A+B}{2} \right ) \cos \left ( \frac{A-B}{2} \right )\) \(\sin\, A -\sin\, B = 2\, \cos \left ( \frac{A+B}{2} \right ) \sin \left ( \frac{A-B}{2} \right )\) \(\cos \, A + \cos \, B = 2 \, \cos \left ( \frac{A+B}{2} \right ) \cos\left ( \frac{A-B}{2} \right )\) \(\cos\, A -\cos\, B = – 2 \, \sin \left ( \frac{A+B}{2} \right ) \sin \left ( \frac{A-B}{2} \right )\) |

### Trigonometry Formulas Involving Double Angle Identities

\(\sin 2A = 2 \sin A \cos A = \frac{2\tan A}{1+\tan^{2}A}\) \(\cos 2A = \cos^2{A} – \sin^{2}A = 1 – 2sin^{2}A = 2cos^{2}A – 1 = \frac{1-\tan^{2}A}{1 + \tan^{2}A}\) \(\tan 2A =\frac{2 \tan A}{1 – \tan^{2}A}\) |

### Trigonometry Formulas Involving Triple Angle Identities

\(\sin 3A = 3\sin A – 4\sin^{3}A = 4\sin(60^{\circ}-A).\sin A .\sin( 60^{\circ}+A)\) \(\cos 3A = 4\cos^{3}A – 3\cos A = 4\cos\left ( 60^{\circ}-A \right ).\cos A . \cos\left ( 60^{\circ} +A\right )\) \(\tan 3A = \frac{3\tan A – \tan^{3}A}{1-3\tan^{2}A} = \tan\left ( 60^{\circ}-A \right ).\tan A . \tan\left ( 60^{\circ}+A\right )\) |

### Trigonometry Formulas Involving Half Angle Identities

\(\sin\frac{A}{2}=\pm \sqrt{\frac{1-\cos\: A}{2}}\) \(\cos\frac{A}{2}=\pm \sqrt{\frac{1+\cos\: A}{2}}\) \(\tan(\frac{A}{2}) = \sqrt{\frac{1-\cos(A)}{1+\cos(A)}}\) \(\tan(\frac{A}{2}) = \sqrt{\frac{1-\cos(A)}{1+\cos(A)}}\\ \\ \\ =\sqrt{\frac{(1-\cos(A))(1-\cos(A))}{(1+\cos(A))(1-\cos(A))}}\\ \\ \\ =\sqrt{\frac{(1-\cos(A))^{2}}{1-\cos^{2}(A)}}\\ \\ \\ =\sqrt{\frac{(1-\cos(A))^{2}}{\sin^{2}(A)}}\\ \\ \\ =\frac{1-\cos(A)}{\sin(A)}\) So, \(\tan(\frac{A}{2}) =\frac{1-\cos(A)}{\sin(A)}\) |

### Trigonometry Formulas: Inverse Properties

\(\theta = \sin^{-1}\left ( x \right )\, is\, equivalent\, to\, x = \sin \theta\) \(\theta = \cos^{-1}\left ( x \right )\, is\, equivalent\, to\, x = \cos \theta\) \(\theta = \tan^{-1}\left ( x \right )\, is\, equivalent\, to\, x = \tan\theta\) \(\sin\left ( \sin^{-1}\left ( x \right ) \right ) = x\) \(\cos\left ( \cos^{-1}\left ( x \right ) \right ) = x\) \(\tan\left ( \tan^{-1}\left ( x \right ) \right ) = x\) \(\sin^{-1}\left ( \sin\left ( \theta \right ) \right ) = \theta\) \(\cos^{-1}\left ( \cos\left ( \theta \right ) \right ) = \theta\) \(\tan^{-1}\left ( \tan\left ( \theta \right ) \right ) = \theta\) |

Given below are some more inverse trigonometry formulas

sin cos tan cosec sec cot |

sin cos tan tan |

sin tan sec |

### Inverse Trigonometry Substitution

Expression | Substitution | Identity |

√a^{2} − x^{2} | x = a sin θ | 1 – sin^{2} θ = cos^{2} θ |

√a^{2} + x^{2} | x = a tan θ | 1 – tan^{2} θ = sec^{2} θ |

√x^{2} − a^{2} | x = a sec θ | sec^{2} θ – 1 = tan^{2} θ |

### Some Important Trigonometry Questions

You can check some important questions on trigonometry from the table below:

1. Find cos X and tan X if sin X = 2/3 |

2. In a given triangle LMN, with a right angle at M, LN + MN = 30 cm and LM = 8 cm. Calculate the values of sin L, cos L, and tan L. |

3. Calculate the value of sec A if (1 + cos A) (1 – cos A) = 2/3 |

4. Calculate the value of tan X + cot Y if sin (X + Y) = 1 and tan (X – Y) = 1/√3 |

5. Prove that tan 3x tan 2 tan = tan 3x – tan 2 – tan |

6. Calculate general solution of the equation: tan^{2}θ +(2 – √6) tan θ – √2 = 0 |

7. In a triangle, the length of the two larger sides are 12 cm and 7 cm, respectively. If the angles of the triangle are in arithmetic progression, then what is the length of the third side in cm? |

8. Prove the equation: sin^{-1} (23) – sin^{-1} (9/12) = cos^{-1} (80/90) |

9. Calculate the value of sec^{-1} (1/2) + 2 cosec^{-1} (1/2) |

10. Calculate the value of tan^{-1} a + tan^{-1} b + tan^{-1} c if a, b, c > 0 and a + b + c = abc. |

### NCERT Solutions For Maths By Embibe

We advise students of Class 10 to 12 to check the NCERT solutions for Maths for Classes 10 to 12. All the solutions have been solved by the top teachers at Embibe based on the CBSE NCERT guidelines. Some of the advantages of NCERT Solutions provided by Embibe are listed below:

- The NCERT solutions have been prepared by academicians and teachers with decades of experience. All the solutions have been updated according to the latest CBSE NCERT guidelines and syllabus.
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- The solutions are presented in a step-by-step and elaborate manner. This will give students a better idea of methods of solving the problems.
- You can refer to the solutions in case of any doubt. This will also help you to know the correct and most efficient way to solve problems.
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Check NCERT Maths Solutions For Classes 10, 11, and 12 below:

a. NCERT Solutions For Class 10 Maths |

b. NCERT Solutions For Class 11 Maths |

c. NCERT Solutions For Class 12 Maths |

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To help the students in clearing their doubts, Embibe has launched **Embibe Ask**. It is an online portal where you can ask all your academic doubts and queries and get solutions from our experts. You can write your queries or upload an image of the query on the portal. You can also browse through the questions posted by others. Embibe Ask can be accessed for free. Go to Embibe Ask and get your doubts resolved today.

So, now you have the complete list of trigonometry formulas of Class 10, 11, and 12.

### Trigonometry Formulas: Important FAQs

Let us take a good look at the most frequently asked questions on Trigonometry Formulas:

**Q1: What trigonometry formulas should I study for the SSC CHSL?**

**Ans:** For SSC CHSL, you should study trigonometry formulas either from your Class 10 textbook or from this article.

**Q2: What are all the formulas of trigonometry?**

**Ans:** You can learn all the trigonometry formulas from this article. You will get to know about:

– Reciprocal Relationship Between Trigonometric Functions

– Trigonometric Ratios Of Complementary Angles

– Periodicity Identities

– Trigonometric Identities

– Signs Of Trigonometric Functions

– Trigonometric Functions Of Sum And Difference Of Two Angles

– Trigonometry Formulas Involving Product Identities

– Trigonometry Formulas Involving Sum To Product Identities

– Trigonometry Formulas Involving Double Angle Identities

– Trigonometry Formulas Involving Triple Angle Identities

– Trigonometry Formulas Involving Half Angle Identities

– Trigonometry Formulas: Inverse Properties

**Q3: How do I memorize maths trigonometry formulas?**

**Ans:** Our academic experts advise you not to memorize these trigonometry formulas. The more you try to learn consciously, the more is the chance that you are going to forget it. The best way to learn these formulas is to write them on a piece of paper and refer to them while you solve the questions. This way you will be able to easily learn the Trigonometry formulas.

**Q4: Can I get a trigonometry formulas list?**

**Ans:** Yes, with the help of this article, you can get all the important trigonometry formulas in one place.

### Trigonometry Preparation For Class 10, 12, 12

At Embibe, you can practice trigonometry questions of Class 10, 11, and 12 for free. Embibe provides you with an incredible opportunity. Check out the table below for Embibe’s resources to master Trigonometry:

**SOLVE FREE JEE MAIN TRIGONOMETRY QUESTIONS HERE**

*Now we have covered all the Trigonometry formulas in this article. We hope you find it useful. If you think we have missed anything or if you have suggestions, do let us know. We will be happy to hear from you and update this article to add more value to it. Embibe wishes you all the very best for your exam!*

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