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Trigonometry formulas are essential mathematical tools that deal with the connections between triangle angles and sides. These formulas, which include sine, cosine, tangent, and their inverses, allow us to handle a variety of real-world distance, height, and angle issues. Understanding and navigating the world around us is made easier by mastering trigonometric formulae.

## List of Important Trigonometry Formulas

Let’s explore trigonometry formulas. Imagine you have a right-angled triangle. This triangle has an angle, which we’ll call θ, a longest side called the hypotenuse, a side that’s right next to θ called the adjacent side, and another side that’s across from θ, which we’ll call the opposite side.

Basic Trigonometric Ratios

- sinθ = Opposite side / Hypotenuse
- cosθ = Adjacent side / Hypotenuse
- tanθ = Opposite side / Adjacent Side
- secθ = Hypotenuse / Adjacent side
- cosecθ = Hypotenuse / Opposite side
- cotθ = Adjacent Side / Opposite side

## Trigonometric Identities

**Trigonometric identities** are special equations that connect various trigonometric functions like sine, cosine, and tangent. They help simplify expressions and solve trigonometric problems.

Reciprocal identities

- sinθ = 1/cosecθ
- cosecθ = 1/sinθ
- cosθ = 1/secθ
- secθ = 1/cosθ
- tanθ = 1/cotθ
- cotθ = 1/tanθ

### Trigonometry Identities

#### Tangent and Cotangent Identities

- tanθ = sin θ / cos θ
- cotθ = cos θ / sin θ

#### Reciprocal Identities

- sinθ = 1/cosecθ
- cosecθ = 1/sinθ
- cosθ = 1/secθ
- secθ = 1/cosθ
- tanθ = 1/cotθ
- cotθ = 1/tanθ

#### Pythagorean Identities

- sin2θ + cos2θ = 1
- 1 + tan2θ = sec2θ
- 1 + cot2θ = cosec2θ

#### Even and Odd Angle Formulas

- sin(-θ) = -sinθ
- cos(-θ) = cosθ
- tan(-θ) = -tanθ
- cot(-θ) = -cotθ
- sec(-θ) = secθ
- cosec(-θ) = -cosecθ

#### Co-function Formulas

- sin(900-θ) = cosθ
- cos(900-θ) = sinθ
- tan(900-θ) = cotθ
- cot(900-θ) = tanθ
- sec(900-θ) = cosecθ
- cosec(900-θ) = secθ

#### Double Angle Formulas

- sin2θ = 2 sinθ cosθ
- cos2θ = 1 – 2sin2θ
- tan2θ = 2 tanθ / 1-tan2θ

#### Half Angle Formulas

- sinθ = ±√1-cos2θ/2
- cosθ = ±√1+cos2θ/2
- tanθ = ±√1+cos2θ/1-cos2θ

#### Thrice of Angle Formulas

- sin3θ = 3sinθ – 4 sin3θ
- Cos 3θ = 4cos3θ – 3 cosθ
- Tan 3θ = 3tanθ – tan3θ/ 1- 3tan2θ
- Cot 3θ = cot3θ – 3cotθ/3cot2θ – 1

#### Product to Sum Formulas

- Sin A Sin B = 1/2 [Cos (A-B) – Cos (A+B)]
- Cos A Cos B = 1/2 [Cos (A-B) + Cos (A+B)]
- Sin A Cos B = 1/2 [Sin (A+B) + Sin (A-B)]
- Cos A Sin B = 1/2 [Sin (A+B) – Sin (A-B)]

#### Sum to Product Formulas

- Sin A + Sin B = 2 sin (A+B)/2 cos (A-B)/2
- Sin A – Sin B = 2 sin (A+B)/2 sin (A-B)/2
- Cos A + Cos B = 2 cos (A+B)/2 cos (A-B)/2
- Cos A – Cos B = 2 cos (A+B)/2 cos (A-B)/2

### Trigonometry table

The table below contains trigonometric formulas for angles that are often utilised in problem solving.

Degrees |
0° |
30° |
45° |
60° |
90° |
180° |
270° |
360° |

Radians |
0 |
π/6 |
π/4 |
π/3 |
π/2 |
π |
3π/2 |
2π |

Sin θ |
0 | 1/2 | 1/√2 | √3/2 | 1 | 0 | -1 | 0 |

Cos θ |
1 | √3/2 | 1/√2 | 1/2 | 0 | -1 | 0 | 1 |

Tan θ |
0 | 1/√3 | 1 | √3 | ∞ | 0 | ∞ | 0 |

Cot θ |
∞ | √3 | 1 | 1/√3 | 0 | ∞ | 0 | ∞ |

Sec θ |
1 | 2/√3 | √2 | 2 | ∞ | -1 | ∞ | 1 |

Cosec θ |
∞ | 2 | √2 | 2/√3 | 1 | ∞ | -1 | ∞ |

### Inverse Trigonometric Functions

If Sin θ = x, then θ = sin-1 x = arcsin(x)

Similarly,

- θ = cos-1x = arccos(x)
- θ = tan-1 x = arctan(x)

Also, the inverse properties could be defined as;

- sin
^{-1}(sin θ) = θ - cos
^{-1}(cos θ) = θ - tan
^{-1}(tan θ) = θ

### Signs of Trigonometric Ratios

- All Six trigonometric ratios are positive in the first quadrant
- Only are positive and the remaining all are negative in the second quadrant.
- Only are positive and the remaining all are negative in the third quadrant.
- Only are positive and the remaining all are negative in the fourth quadrant.

## FAQs on Trigonometric Formulas

### What is the basic formula for trigonometry

The basic formula in trigonometry relates the angles and sides of a right-angled triangle, such as sine, cosine, and tangent.

### What are the 7 formula of trigonometry?

There are 7 fundamental trigonometric formulas, including sine, cosine, tangent, cosecant, secant, cotangent, and the Pythagorean theorem.

### What is 15 trigonometry formula?

There is no standard 15 trigonometry formula. The core trigonometric formulas are usually limited to the fundamental 7.

### What is the trigonometry formula for Class 11?

Trigonometry formulas for Class 11 cover the basics, including sine, cosine, tangent, and their inverses, as well as trigonometric identities.

### Who is the father of trigonometry?

The father of trigonometry is considered to be Hipparchus.

### Who is the mother of trigonometry?

There isn't a widely recognized mother of trigonometry as it is a collective field of mathematics.

### What is theta angle?

Theta (θ) is a symbol used in trigonometry to represent an angle.

### What are the 4 types of trigonometry?

The four main types of trigonometry are plane trigonometry, spherical trigonometry, hyperbolic trigonometry, and analytic trigonometry.