# Trigonometry

** Trigonometry** is a branch of mathematics that explores the relationships between the

**of a**

**ratios of the sides****and its angles. Trigonometry word is formed from ancient Greek words “**

**right-angled triangle****” and “**

**trigonon****” which mean**

**metron****respectively, thus collectively called Trigonometry which means measures of a triangle.**

**triangle and measure**The fundamental ratios used to study these relationships are known as ** trigonometric ratios**, which include sine, cosine, tangent, cotangent, secant, and cosecant. The term “trigonometry” is a 16th-century Latin derivative and the concept was given by the Greek mathematician Hipparchus.

The most important topics in trigonometry are **trigonometry table**** , trigonometry formulas, trigonometric identities, and trigonometric ratios.** In the following sections, we will see the

**, including its**

**basics of trigonometry****and**

**fundamental identities****.**

**formulas**Table of Content

## What is Trigonometry?

Trigonometry is the branch of mathematics that deals with the side and angles of a triangle, especially the right angle triangle i.e., a triangle with one angle right angle. Other than right angles, trigonometry is helpful in many different geometric figures, either 2-dimensional or 3-dimensional. Studies of Trigonometry can be classified into three parts which are as follows:

(deal with right angle triangles only)**Core Trigonometry**(deals with all types of 2-dimensional geometry)**Plane Trigonometry**(deals with all types of 3-dimension geometry)**Spherical Trigonometry**

## Trigonometry Basics

Trigonometry Basics are the core concepts of trigonometry without which it can’t be defined, some of these basics of trigonometry are as follows:

The measure of space between two intersecting lines are known as angles.**Angles:**A triangle with one of its interior angles being the right angle i.e., 90Â°, is called right angles triangle.**Right-angle Triangle:**In right angles triangle, according to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of squares of the other two sides,**Pythagoras Theorem:**Trigonometric Ratios are defined as the ratio of the sides of the right angle triangles. As there are 3 ways to choose two sides out of three and two ways for each chosen pair to arrange in ratio, thus there are 3Ã—2 =6 trigonometric ratios which are defined for each possible pair of sides of the right angle triangle.**Trigonometric Ratios:**

## Important Trigonometry Functions

The important trigonometric functions include sin and cos, as all the other trigonometric ratios can be defined in terms of sin and cos. The six important trigonometric functions (trigonometric ratios) are calculated using the below formulas.

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 |

## Even and Odd Trigonometric Functions

** Odd trigonometric functions:** A trigonometric function is said to be an odd function if f(-x) = -f(x) and symmetric with respect to the origin.

** Even trigonometric functions:** A trigonometric function is said to be an even function, if f(-x) = f(x) and symmetric to the y-axis.

- Sin (-x) = â€“ Sin x
- Cos (-x) = Cos x
- Tan (-x) = -Tan x
- Csc (-x) = â€“ Csc x
- Sec (-x) = Sec x
- Cot (-x) = -Cot x

## Trigonometric Ratios

In the study of Trigonometry, there are six trigonometric ratios i.e., sin, cos, tan, cot, sec, and cosec. These trigonometric ratios as the name suggest defined as the ratios between two sides of a right-angle triangle. For the sin trigonometric ratio, we defined it as the ratio of the perpendicular (opposite to the angle for which we are calculating sin,) to the hypotenuse of the right-angle triangle i.e., for triangle ABC right angles at B,

Similarly, cos is defined as the ratio of the base and hypotenuse of the right-angle triangle,

and, trigonometric ratio tan is defined as the ratio of perpendicular to the base of the right-angle triangle.

Other, then sin, cos and tan, cot, sec, and cosec are also defined as the ratio of the sides of right angle triangle as follows:

** Note: **sec, cosec, and cot are the reciprocals of the trigonometric ratios cos, sin, and tan respectively.

## Trigonometry Table

Trigonometric Table is the table of values of trigonometric ratio at some standard angle values. The table for the value of trigonometric ratios at different angles is given below:

## Trigonometry Chart

To create a chart of different values of trigonometric ratios at different angles, we can use the following algorithm. Using this trick we can complete the trigonometry chart without remembering the exact values of the ratios for different angles. The steps required in this algorithm are as follows:

Write first five whole numbers with some distance between them.Step 1:

0 | 1 | 2 | 3 | 4

Divide each number by 4.Step 2:

0 | 1/4 | 2/4 | 3/4 | 4/4OR

0 | 1/4 | 1/2 | 3/4 | 1

Take Square Root for each resulting number in step 2..Step 3:

âˆš0 | âˆš(1/4) | âˆš(1/2) | âˆš(3/4) | âˆš1OR

0 | 1/2 | 1/âˆš2 | âˆš(3)/2 | 1

Resulting values for step 3, are the value of trigonometric ratio sine for angles 0Â°, 30Â°, 45Â°, 60Â°, 90Â°.Step 4:

Reverse the order of the resulting values in step 3, to get the value of cos for the same angles.Step 5:

1 | âˆš(3)/2 | 1/âˆš2 | 1/2 | 0

Find the ratio of the results in step 3 to results in step 5, to get the value of cos for angles 0Â°, 30Â°, 45Â°, 60Â°, 90Â°.Step 6:

0Ã·1 | 1/2Ã·âˆš(3)/2 | 1/âˆš2Ã·1/âˆš2 | âˆš(3)/2Ã·1/2 | 1Ã·0OR

0 | 1/âˆš3 | 1 | âˆš(3) | Not definedThus, using these steps, resulting table is formed.

Angle

0Â°

30Â°

45Â°

60Â°

90Â°

sin

0

1/2

1/âˆš2

âˆš(3)/2

1

cos

1

âˆš(3)/2

1/âˆš2

1/2

0

tan

0

1/âˆš3

1

âˆš3

Not Defined

## Trigonometry Angles

The angle for which the trigonometric ratio is defined is the trigonometric angle. Angles can either be measured in degrees (Â°) or can be measured in radians (rad). Some standard angles for which we create the table of trigonometric ratios are 0Â°, 30Â°, 45Â°, 60Â°, and 90Â°. Other than these angles, we also sometimes need to deal with 15Â°, 18Â°, 75Â°, and 72Â°.

## Application of Trigonometry

Trigonometry has so many applications in the real world, we can even say that is the most used mathematics concept throughout mathematics. Some of the applications of trigonometry are as follows:

- Trigonometry is very essential for modern-day
such as GPS or any other similar system.**navigation systems** - In most streams of
trigonometry is used extensively for various kinds of analysis and calculations, which helps engineers to make more sound decisions for the construction of various kinds of structures.**engineering,** - Various trigonometric concepts and formulas are used in the
of the modern age, as computer graphics are created in 3-D environments so all the calculations are done by the graphics processing unit of the computer to deliver the computer graphics as output.**computer graphics** - Various
such as the radius of celestial bodies, the distance between objects, etc. involve the use of trigonometry and its different trigonometric ratios.**astronomical calculations** - In
, we use trigonometry to understand and evaluate many real-world systems such as the orbits of planets and artificial satellites, the reflection or refraction of light ni various environments, etc.**Physics**

## Trigonometry Formulas List

In trigonometry, there are a lot of formulas or identities that relate the different trigonometric ratios with each other for different values of angles. Some of these identities are:

**Pythagorean Trigonometric Identities**

**Pythagorean Trigonometric Identities**

- sin
^{2}Î¸ + cos^{2}Î¸ = 1- 1+tan
^{2}Î¸ = sec2 Î¸- cosec
^{2}Î¸ = 1 + cot^{2}Î¸

**Sum and Difference Identities**

**Sum and Difference Identities**

- 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) = (tan A + tan B)/(1 – tan A tan B)
- tan (A-B) = (tan A – tan B)/(1 + tan A tan B)

** For more**:- Sum and Difference Identities

**Double angle Identities**

**Double angle Identities**

- sin 2Î¸ = 2 sinÎ¸ cosÎ¸
- cos 2Î¸ = cos
^{2}Î¸ – sin^{2}Î¸ = 2 cos^{2 }Î¸ – 1 = 1 – sin^{2}Î¸- tan 2Î¸ = (2tanÎ¸)/(1 – tan
^{2}Î¸)

**Half Angle Identities**

**Half Angle Identities**

**Product Sum Identities**

**Product Sum Identities**

**Product Identities**

**Product Identities**

**Triple Angle Identities**

**Triple Angle Identities**

## Trigonometry Identities

An equation, which deals with different trigonometric ratios is called Trigonometric Identity if it is true for all possible values of the angles. Trigonometric identities related different trigonometric ratios i.e., sin, cos, tan, cot, sec, and cosec, with each other for various different angles. Out of all one of the most basic as well as useful identities are Pythagorean trigonometric identities which are given as follows:

sin^{2}Î¸ + cos^{2}Î¸ = 1

1+tan^{2}Î¸ = sec2 Î¸

cosec^{2}Î¸ = 1 + cot^{2}Î¸

## Eulerâ€™s Formula for Trigonometry

For the imaginary power of exponent e(Euler’s number), Eular gave an identity that relates the imaginary power of e to the trigonometric ratios sin and cos, the identity is given as follows:

Where,

is the imaginary number which is defined asi, andi = âˆš(-1)

is the angle.Î¦Putting,

instead of-Î¦in the above identity, we getÎ¦Now, adding and subtracting these two values together we get, values of sin and cos in terms of imaginary power of Eular’s number,

and

## Trigonometry Real-Life Example

As Trigonometry is the most widely used concept in mathematics, it has various applications and examples in the real world. One such example is height and distance. In Height and distance, we can calculate various lengths and angles involving everyday scenarios. For example, we see a shadow of a tree on the ground and want to find the height of that tree (which is very hard to measure as we have to reach to the top of the tree to measure its height). Using trigonometry we can calculate the height of the tree without climbing the tree at all. we just need to measure the angle of the sun at the moment and the length of the shadow of the tree at the same moment and using this information we can use trigonometric ratio tan to calculate the height of the tree.

## Trigonometry Problems and Solutions

**Example 1: A ladder is leaning against a wall. The angle between the ladder and the ground is 45 degrees, and the length of the ladder is 10 meters. How far is the ladder from the wall?**

**Solution:**

Let the distance between the ladder and the wall be x meters.

Here, ladder, wall and ground together makes a right angle triangle, where for given angle,

Length of ladder = hypotenous = 10 meter,

Distance between wall and laddar = base = x meterUsing trigonometric ratio cos, we get

â‡’ cos(45Â°) = = x/10

â‡’ cos(45Â°) = 1

â‡’1/âˆš2 = x/10

â‡’ x = 10/âˆš2 = 5âˆš2 meters

Therefore, the ladder is 5âˆš2 meters away from the wall.

**Example 2: A right-angled triangle has a hypotenuse of length 10 cm and one of its acute angles measures 30Â°. What are the lengths of the other two sides?**

**Example 2: A right-angled triangle has a hypotenuse of length 10 cm and one of its acute angles measures 30Â°. What are the lengths of the other two sides?**

**Solution:**

Let’s call the side opposite to the 30Â° angle as ‘a’ and the side adjacent to it as ‘b’.

Now, sin (30Â°) = perpendicular/hypotenous = a/10

â‡’ a = 10 Ã— sin(30Â°) = 5 cm [sin(30Â°) = 1/2]

and cos(30Â°) = b/10

â‡’ b = 10 Ã— cos(30Â°) = 10 Ã— âˆš(3)/2 â‰ˆ 8.66 cm

Therefore, the lengths of the other two sides are 5 cm and 8.66 cm (approx.).

**Example 3: Prove that (cos x/sin x) + (sin x/cos x) = sec x Ã— cosec x.**

**Example 3: Prove that (cos x/sin x) + (sin x/cos x) = sec x Ã— cosec x.**

**Solution:**

LHS = (cos x/sin x) + (sin x/cos x)

â‡’ LHS = [cos^{2}x + sin^{2}x]/(cos x sin x)

â‡’ LHS = 1/(cosx sinx) [Using cos^{2}x + sin^{2}x = 1]

â‡’ LHS = (1/cosx) Ã— (1/sinx)

â‡’ LHS = secx Ã— cosecx = RHS [ 1/cosx = sec x and 1/sinx = cosec x]

**Example 4: A person is standing at a distance of 10 meters from the base of a building. The person measures the angle of elevation to the top of the building as 60Â°. What is the height of the building?**

**Example 4: A person is standing at a distance of 10 meters from the base of a building. The person measures the angle of elevation to the top of the building as 60Â°. What is the height of the building?**

**Solution:**

Let h be the height of the building.

and, all the distances here in the question make a right angle triangle, with a base of 10 meters and height h meter.

As tan Î¸ = Perpendicular/Base

â‡’ tan(60Â°) = h / 10

â‡’ h = 10 tan(60Â°)

Using the values of tan(60Â°) = âˆš3, we get:

h = 10âˆš3 â‰ˆ 17.32 m

Therefore, the height of the building is approximately 17.32 meters.

**Example 5: Find the value of x in the equation cos**^{-1}**(x) + sin**^{-1}**(x) = Ï€/4.**

**Example 5: Find the value of x in the equation cos**

^{-1}

**(x) + sin**

^{-1}

**(x) = Ï€/4.**

**Solution:**

For, cos

^{-1}(x) + sin^{-1}(x) = Ï€/4As we know, cos

^{-1}(x) + sin^{-1}(x) = Ï€/2, above equation becomesÏ€/2 = Ï€/4, which is not true.

Thus, the given equation has no such value of x, which can satisfy the equation.

**Also Check:**

- What are the six trigonometry functions?
- Trigonometry Formulas
- Trigonometric Ratios
- Trigonometry Table
- Applications of Trigonometry
- List of All Trigonometric Identities
- Introduction to Trigonometry Class 10 Maths Notes Chapter 8
- Trigonometric Equations

## FAQs – Trigonometry

### 1. What is Trigonometry in Maths?

Trigonometry is the branch of mathematics that deals with the relationship between angles and lengths in geometric shapes.

### 2. What are the three types of trigonometry?

The three types of trigonmetry are as follows:

- Core Trigonometry
- Plane Trigonometry
- Spherical Trigonometry

### 3. What are the Six ratios of trigonometry?

The six trigonometric ratios are

andsin, cos, tan, cot, sec,cosec.

### 4. What is the primary function of trigonometry?

The primary function of the trigonometry is to study and understand the relationships between sides and angles of triangles.

### 5. What are Trigonometry Identities?

An equation that holds true for all angles involving different trigonometric ratios is known as trigonometric identity.

### 6. Who Invented Trigonometry?

Trigonometry as a concept, was first introduced by an ancient Greek mathematician Hipparchus.

### 7. What are the Basics of Trigonometry?

The right angle triangle is the most basic thing in trigonometry as all the trigonometric ratios are defined as the ratio of sides of right angle triangles.

### 8. Where trigonometry is used in real life?

Trigonometry and its functions find diverse applications in our everyday lives. It’s instrumental in geography for measuring distances between landmarks, in astronomy for gauging distances to nearby stars, and crucial in satellite navigation systems for accurate positioning and mapping.

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