# Trigonometry

The term Trigonometry can be described in the following points:

**Trigonometry**is part of math that looks at how the**lengths of the sides of a right triangle**relate to the**angles**.**Trigonometric ratios**are used to figure out how this relationship works.- These ratios are
**sine**,**cosine**,**tangent**,**cotangent**,**secant**, and**cosecant**. - The word “
**trigonometry**” comes from**Latin**, and**Hipparchus**, a Greek mathematician, invented the idea.

## What is Trigonometry

The word “**trigonometry**” is a **Greek word**. Its means “**measurement of a triangle**.” Therefore trigonometry is that branch of mathematics concerned with measuring the **sides **and **angles **of a **plane triangle** and investigating the various relations among them. Today the subject of trigonometry also includes another distinct branch that concerns itself with **properties relations** between and behavior of **trigonometric functions**. The importance of trigonometry will be immediately realized when its applications in solving the problem of mensuration, mechanics physics, surveying, and astronomy are encountered.

## Trigonometry Definition

## Elements of Trigonometry

There are some important elements of trigonometry which are as follows;

**Angles****Quadrants**

### Angles

OR

In the figure, the first position OX is the initial line, and the second position, OP, is the terminal line or generating line of <XOP.

If the terminal side resolves in an **anticlockwise direction**, the angle described is positive, as shown in the figure.

If the terminal side resolves in a **clockwise direction**, the angle describes is negative, as shown in the figure

### Quadrants

Thus XOY, X’OY, X’OY’, and XOY’ are called the 1st, 2nd, 3rd, and 4th quadrants, respectively.

**1st Quadrant**

In the first quadrant, the angle varies from**0° to 90°**in an**anti-clockwise**direction and from –**270° to – 360°**in a**clockwise**direction.**2nd Quadrant**

In the second quadrant, the angle varies from**90° to 180°**in an**anti-clockwise**direction and –**180° to 270 °**in a**clockwise**direction.**3rd Quadrant**

In the third quadrant, the angle varies from**180° to 270°**in an**anti-clockwise**direction and from**– 90° to-180°**in a**clockwise**direction.**4th Quadrant**

In the fourth quadrant, the angle varies from**270° to 360°**in an**anti-clockwise**direction and from**– 0° to-90°**in a**clockwise**direction.

## Types of Trigonometry

There are two types of Trigonometry;

**Plane Trigonometry****Spherical Trigonometry**

**Plane Trigonometry**

**Spherical Trigonometry**

## Trigonometric Ratios

In trigonometry, **six fundamental ratios** help us to find the **relationship **between the **lengths **of the sides of a **right triangle** and the **angle**. Let’s look at how these ratios or functions work in the case of a triangle with a right angle. Think about a **triangle **with a **right angle**. The longest side is called the **hypotenuse**, and the sides next to and opposite the hypotenuse are called **adjacent and opposite sides**.

If the angle between the **base **and the **hypotenuse **of a triangle with a right angle **θ**, then;

**sin θ**= Perpendicular/Hypotenuse**cos θ**= Base/Hypotenuse**tan θ**= Perpendicular/Base**tan θ**=**sin θ**/**cos θ**

**Cot θ**= Base/Perpendicular**Cot θ**=**cos θ**/**sin θ**

**Sec θ**= Hypotenuse/Perpendicular**Cosec θ**= Hypotenuse/Perpendicular

Functions | Abbreviation | Relationship to sides of a right triangle |

Sine Function | sin | Opposite side/ Hypotenuse |

Tangent Function | tan | Opposite side / Adjacent side |

Cosine Function | cos | Adjacent side / Hypotenuse |

Cosecant Function | cosec | Hypotenuse / Opposite side |

Secant Function | sec | Hypotenuse / Adjacent side |

Cotangent Function | cot | Adjacent side / Opposite side |

### Trigonometric Reciprocal Functions

The reciprocal functions of Trigonometric Ratios are as follows;

sin θ = 1/Cosec θ | Cosec θ = 1/sin θ |

cos θ = 1/Sec θ | Sec θ = 1/cos θ |

tan θ = 1/Cot θ | Cot θ = 1/tan θ |

### Difference b/w Even and Odd Trigonometric Functions

The trigonometric function is either even or odd.

Odd Trigonometric Functions | Even Trigonometric Functions |

If f(-x) = -f(x) and the function is symmetric from the origin, it is called an odd function. | A trigonometric function is said to be even if f(-x) = f(x) and the position is the same on both sides of the y-axis. |

Sin (-x) = – Sin xTan (-x) = -Tan x Cosec (-x) = – Cosec x Cot (-x) = -Cot x | Cos (-x) = Cos xSec (-x) = Sec x |

## Trigonometric Ratios of Particular Angles

**Trigonometric angles** are the angles in a triangle with a right angle that can be used to show different **trigonometric functions**. The angles** 0 ^{o}, 30^{o}, 45^{o}, 60^{o}, and 90^{o}** are often used in trigonometry. A trigonometric table shows the exact trigonometric values for each of these angles.

**180**are also important angles in trigonometry. In terms of trigonometric ratios, an angle in trigonometry can be written as,

^{o}, 270^{o}, and 360^{o}**θ = sin**(Perpendicular/Hypotenuse)^{-1}**θ = cos**(Base/Hypotenuse)^{-1}**θ = tan**(Perpendicular/Base)^{-1}

### Trigonometric Ratio of 0^{o}

Let the **initial line **revolve and trace a slight angle nearly equal to **zero 0°**. Take a **point P** on the **final line**.

Draw **PM** perpendicularly on **OX**.**PM = 0**

### Trigonometric Ratio of 30^{o} or π/6

Let the **initial line OX **revolve and trace out an **angle of 30°**. Take a point **P on the final line**. Draw **PQ** **perpendicular **to **P on OX**. In a 30° right-angled triangle, the side opposite to the 30° angle is **one-half** the length of the **hypotenuse**, i.e., if **PQ** = **1 unit**, then **OP** will be **2 units**.

From fig. **OPQ **is a** right-angled triangle**

By the **Pythagorean theorem**, we have**(OP)² = (OQ)² + (PQ)²(2)² = (0Q)² + (1)²4= (0Q)² + 1(0Q)² = 3(0Q) = √3**

### Trigonometric Ratio of 45^{o} or π/4

Let the **initial line OX** revolve and trace out an **angle of 45°**. Take a **point P** on the **final line**. Draw PQ perpendicular to **P** on **OX**. In a** 45° right-angled triangle**, the perpendicular length is equal to the length of the **base**, i.e., if** PQ = 1 unit** then **OQ = 1 unit**.

From figure by **Pythagorean theorem**.**(OP)² = (OQ)² + (PQ)²(OP)² = (1)² + (1)² = 1+1=2OP = √2**

### Trigonometric Ratio of 60^{o} or π/3

Let the **initial line OX** revolve and trace out an angle of** 60°**. Take a **point P **on the **final line**. Draw **PQ** perpendicular to **P on OX**. In a **60° right angle triangle**, the length of the Base is **one-half **of the **Hypotenuse** i.e., **OQ = Base = 1 unit**, then. **OP Hyp. = 2 units**

from a figure by **Pythagorean Theorem**:**(OP)² = (OQ)² + (PQ)²(2)² = (1)² + (PQ)²4 = 1 (PQ)²(PQ)² = 3PQ = √3**

### Trigonometric Ratio of 90^{o}

Let the initial line revolve and trace out an angle nearly **equal to 90°**. Take a **point P** on the **final line**. Draw** PQ** perpendicular to **P on OX**.**OQ=0**, **OP = 1**, **PQ = 1** (Because they coincide y-axis).

## Trigonometry Table

Angles | 0° | 30° | 45° | 60° | 90° |

Sin θ | 0 | ½ | 1/√2 | √3/2 | 1 |

Cos θ | 1 | √3/2 | 1/√2 | ½ | 0 |

Tan θ | 0 | 1/√3 | 1 | √3 | ∞ |

Cosec θ | ∞ | 2 | √2 | 2/√3 | 1 |

Sec θ | 1 | 2/√3 | √2 | 2 | ∞ |

Cot θ | ∞ | √3 | 1 | 1/√3 | 0 |

## Signs of Trigonometric Functions

The trigonometric ratios have **different signs** in **different quadrants**. We can remember the character of the trigonometric function by the “**ACTS**” or **CAST **rule. In “CAST,” **C** stands for **cosine**, **A** stands for **All**, **S** stands for **Sine**, and** T **stands for **Tangent**.

**First Quadrant**In the first quadrant, the sign of all the trigonometric functions is**positive**, i.e., Sin, cos, tan, Cot, Sec, and Cosec are all positive.**Second Quadrant**In the second quadrant,**Sine**and its inverse**cosec**are**positive**. The remaining four trigonometric functions, i.e.,**cos**,**tan**,**cot**,**sec**, are all**negative**.**Third Quadrant**In the third quadrant,**tan**and its reciprocal**cot**are**positive**and the remaining four functions, i.e.,**Sin**,**cos**,**sec**, and**cosec**, are all**negative**.**Fourth Quadrant**In the fourth quadrant,**cos**and its reciprocal**sec**are**positive**; the remaining four functions, i.e.,**Sin**,**tan**,**cot**, and**cosec**, are**negative**.

### Rectangular Co-ordinates and Sign Convention

In plane geometry, the position of a point can be fixed by measuring its perpendicular distance from each of two perpendicular called **coordinate axes**. The **horizontal line (x-axis)** is also called **abscissa**, and the **vertical line(y-axis) **is called **ordinate**.

The distance measured from **point O** in the direction **OX **and **OY **are regarded as **positive**, while in the direction of **OX’ **and **OY’** are considered **negative**. Thus in the given figure **OM _{1}**,

**OM**,

_{4}**MP**and

_{1 }**M₂P**are

_{2}**positive**, while

**OM**,

_{2}**OM**,

_{3}**M**and

_{3}P_{3}**M**are

_{4}P_{4 }**negative**. The terminal line, i.e.,

**OP**,

_{1}**OP**,

_{2}**OP**, and

_{3}**OP**, are

_{4}**positive**in all the

**quadrants**.

## Trigonometry Formulas

Pythagorean Identities | Sum and Difference identities | If A, B, and C are angles and a, b, and c are the sides of a triangle, then, |

1- sin²θ + cos²θ = 12- tan^{2}θ + 1 = sec^{2}θ3- cot^{2}θ + 1 = cosec^{2}θ4- sin 2θ = 2 sin θ cos θ5- cos 2θ = cos²θ – sin²θ6- tan 2θ = 2 tan θ / (1 – tan²θ)7- cot 2θ = (cot²θ – 1) / 2 cot θ | For angles u and v, we have the following relationships:sin(u + v) = sin(u)cos(v) + cos(u)sin(v)cos(u + v) = cos(u)cos(v) – sin(u)sin(v)sin(u – v) = sin(u)cos(v) – cos(u)sin(v)cos(u – v) = cos(u)cos(v) + sin(u)sin(v)The remaining formulas are given below | Sine Laws:a/sinA = b/sinB = c/sinC Cosine Laws:ca^{2 }= ^{2 }+ b^{2 }– 2ab cos Ca b^{2 }=^{2 }+ c^{2 }– 2bc cos Ab a^{2 }=^{2 }+ c^{2 }– 2ac cos B |

**Sum and Difference identities remaining formulas**

## What is Right Angle Trigonometry?

In right-angle trigonometry, sine (sin), cosine (cos), and tangent (tan) are the three fundamental trigonometric ratios. They are defined as follows:

**Sine:**In a right triangle, the ratio of the lengths of the sides directly opposite the angle and the hypotenuse is known as the sine of the arc. Sin is indicated by it (theta).**Cosine:**In a right triangle, the cosine of an angle is the ratio of the angle’s adjacent side’s length to the hypotenuse’s length. It has the symbol cos (theta).**Tangent:**In a right triangle, the angle’s tangent is the proportion of the lengths of its neighboring and opposing sides. It’s designated as tan (theta).

These ratios are employed in several trigonometric equations to determine the sides or unknown angles of a right triangle. Engineering, physics, and navigation are just a few disciplines that use right-angle trigonometry.

## Application of Trigonometry

Trigonometry has been used in many fields, such as **architecture**, **astronomy**, **surveying**, etc. Some ways it can be used are:

**Oceanography**,**seismology**,**meteorology**, the**physical sciences**,**astronomy**,**acoustics**,**navigation**,**electronics**, and many more.- It can also be used to figure out
**how far away long rivers are**,**how tall a mountain is**, etc. - Trigonometry on a sphere has been used to find the
**positions of the sun, moon, and stars**.

## FAQ’s

### Who is the father of trigonometry?

Trigonometry is a branch of math that looks at the relationships between the lengths of the sides of a triangle and the angles. Hipparchus is thought to have invented trigonometry. Hipparchus, a Greek mathematician from the second century BC, was the first to learn how to use trigonometry. He also made the first trigonometric table and solved many problems in spherical trigonometry.

### What is the conclusion for trigonometry?

Its literal meaning is “triangle measurement,” but it is used for much more than just measuring triangles. Trigonometry may not be very useful in everyday life, but it does make it easier to work with triangles. It’s an excellent addition to geometry and accurate measurements, so it’s a good idea to learn the basics even if you don’t want to go further.

### What is the golden formula of trigonometry?

Bisect BC at P; the right triangle ABP has angles 18° at A and 72° at P. We know that AB = τ and BP = 1/2; therefore

sin 18° = cos 72° = 1/(2τ) = (τ – 1)/2 .

From the Pythagorean theorem,

cos² 18° = 1 – sin² 18° = 1 – (τ – 1)²/4

= 1 – (2 – τ)/4

= (2 + τ)/4.

Therefore

cos 18° = sin 72° = (1/2)√(2 + τ).

cos 36° = sin 54° = τ/2 .

sin 36° = cos 54° = (1/2)√(3 – τ).

### Why is tan called tan?

The term “tan” in trigonometry is short for tangent. The word tangent comes from the Latin “tangent”, which means “touching”. In trigonometry, the tangent function represents the ratio of the length of the opposite side to the length of the adjacent side of a right triangle, where the angle between the hypotenuse and the adjacent side is measured.

### What is theta formula?

In math, the Greek letter theta (θ) is often used to represent an angle. Depending on the situation, theta can be used in many different formulas. These are a few examples:

The formulas for sine, cosine, and tangent:

sin(θ) = opposite/hypotenuse

cos(θ) = adjacent/hypotenuse

tan(θ) = opposite/adjacent

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