Trigonometry is the mathematical expression of using special techniques to find distances and altitudes, edges, and angles of inclination within a triangle. Previously, astronomers used trigonometry and other mathematical formulas to estimate the distance of the Earth from celestial bodies such as the sun, moon, and stars. Trigonometry is used to modify complex problems in various natural sciences and conceptual representations in the modern era. Trigonometry was introduced by Hipparchus, a Greek mathematician, who created the first known chord table around 140 BC. His research on spherics had also survived and is the earliest recorded work on spherical trigonometry.

Table of Contents |

Trigonometry |

Trigonometric Ratio |

Trigonometric Sign |

Values of Trigonometric Ratio |

Ratio Formula |

Complementary Ratio |

Supplementary Ratio |

Pythagoras Theorem |

Conclusion |

FAQs |

**Trigonometric Ratio**

A trigonometric ratio is a ratio of the lengths of triangle sides. In trigonometry, these Ratios pertain to the proportion of sides of a triangle to the corresponding angle. The fundamental trigonometric ratios are sin, cos, and tan, or sine, cosine, and tangent ratios. The sin, cos, and tan functional areas can be used to calculate the other essential trig ratios, cosec, sec, and cot. The values of these equivalent fractions can be determined by calculating using the acute angle measure in a right-angled triangle. The acute angle is less than 90° in length. When two rays intersect at a vertex, an angle is formed. When this angle is much less than 90°, this is referred to as an acute angle.

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**Trigonometric Sign**

Trigonometric Sign is determined by the signs of dimensions of the points on the angle’s terminal side. Knowing which quadrant the end side of an angle is in allows for determining the signs of all trigonometric functions. The terminal side can be found in any four quadrants and along the axes, either in the positive or negative path. For the indications of the trigonometric functions, each circumstance means something different. The distance between two points has always been favourable, but the signs of x and y coordinates can be positive or negative.

- sin (-θ) = − sin θ
- cos (−θ) = cos θ
- tan (−θ) = − tan θ
- cosec (−θ) = − cosec θ
- sec (−θ) = sec θ
- cot (−θ) = − cot θ

Ratios |
Values |

sin θ | Perpendicular/ Hypotenuse |

cos θ | Base/ Hypotenuse |

tan θ | Perpendicular/ Base |

cot θ | Hypotenuse/ Base |

sec θ | Hypotenuse/ Perpendicular |

cosec θ | Base/ Perpendicular |

**Values of Trigonometry Ratios**

∠A |
0 Degree |
30 Degree |
45 Degree |
60 Degree |
90 Degree |

sin A | 0 | 1/2 | 1/√2 | √3/2 | 1 |

cos A | 1 | √3/2 | 1/√2 | 1/2 | 0 |

tan A | 0 | 1/√3 | 1 | √3 | – |

cosec A | – | 2 | √2 | 2/√3 | 1 |

sec A | 1 | 2/√3 | √2 | 2 | – |

**Ratios Formula**

- Tan θ = sin θ /cos θ
- Cot θ = cos θ /sin θ
- Sin θ = 1 /cosec θ
- Cos θ = sin θ /tan θ = 1/sec θ
- Sec θ = tan θ /sin θ = 1 /cos θ
- Cosec θ = 1/sin θ

**Reciprocal between Ratios**

Value |
Relation |

Tan A | sin A/cos A |

Cot A | cos A/sin A |

Cosec A | 1/sin A |

Sec A | 1/cos A |

**Complementary Ratio**

A complementary Ratio is defined as the set of two angles whose total value equals 9yo 0°.

If the sum of any two angles equals 900, they are complementary. In other words, the conjunction of any angular position is the value by subtracting this from 900. The total amount of other two angles in a right-angled triangle, excluding the right angle, equals 900. As a result, such angles are considered complementary. Take into account a right triangle ABC right angled at B to generate the trigonometric ratios of a complementary angles formula. If the angle at “C” is chosen to take as the reference angle “, therefore the angle at “A” is indeed the equivalent of the angle at “C.” Hence, the angle at ‘A’ = 900.

- Sin (900 – A) = Cos A
- Cos (900 – A) = Sin A
- Tan (900 – A) = Cot A
- Cot (900 – A) = Tan A
- Sec (900 – A) = Cosec A
- Cosec (900 – A) = Sec A

** ****Supplementary Ratio**

Supplementary Angle Ratio sums up to 180 degrees are regarded as supplementary angles. Angles 130° and 50°, the angles that are complementary to one another do not have to be adjacent. As a result, a certain angle could be supplementary if their sum is equal to 180°.

Only when the sum of two vantage points equals 180° the angles are referred to as supplementary angles. In other words, supplementary angles have been formed when two angles add together to shape a straight angle. These same two angles shape a linear angle, which means that even if each angle is x, the other is 180 – x. The linearity, in this case, proves that perhaps the characteristics of both the angles remain constant.

- sin (180°- θ) = sinθ
- cos (180°- θ) = -cos θ
- cosec (180°- θ) = cosec θ
- sec (180°- θ)= -sec θ
- tan (180°- θ) = -tan θ
- cot (180°- θ) = -cot θ

**Trigonometric Identities**

Trigonometric Identities come in use when trigonometric features are used in an expression and equation. Sine and cosine identities hold for any value of a variable that appears on both sides of the equation. Geometrically, such identities involve one or more transformation functions. The primary trigonometry features are sine, cosine, and tangent, while another three functions are cotangent, secant, and cosecant. All six trigonometric ratio functions are used to derive trigonometric identities. These equivalent fractions are described using the sides of a right triangle, particularly the adjoining, opposite, and hypotenuse ends of the spectrum.

**Reciprocal Identities**

Reciprocal Identities are also the reciprocals of the six primary trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. It is essential to understand that mutual identities are not like inverse trigonometric functions. Every basic trigonometric function is the inverse of another fundamental trigonometric function.

- Sin θ = 1/Csc θ or Csc θ = 1/Sin θ
- Cos θ = 1/Sec θ or Sec θ = 1/Cos θ
- Tan θ = 1/Cot θ or Cot θ = 1/Tan θ

**Pythagorean Identities**

Pythagorean Identities are being used to solve numerous trigonometric problems in which one trigonometric proportion is provided and the other ratios must be determined. The foundational Pythagorean identity expresses the relationship among sin and cos and is the most widely used Pythagorean identity, as follows:

cos2 θ + sin2 θ = 1

tan2 θ – sec2 θ = 1

cot2 θ – cosec2 θ = 1

**Pythagoras Theorem**

Pythagoras’s Theorem describes the relationship between different edges of the right-angled triangle. According to Pythagoras ‘ theorem, the hypotenuse is equivalent to the total of the squares of the other two sides of the triangle. According to Pythagoras’ theorem, when a triangle is right-angled (90 °), the square of the hypotenuse equals the sum of squares of the other two sides.

According to Pythagoras’ theorem, “the square of a hypotenuse side in a right-angled triangle is equal to the sum of the squares of the other two sides.” Perpendicular, Base, and Hypotenuse are the names given to the triangle’s sides. This same hypotenuse is the largest of the three in this case because it is perpendicular to the 90° angle. Pythagoras Theorem is used to determine whether or not the triangle is such a right-angled triangle. If the lengths of the other two sides of the right-angled triangle are known, it is easy to determine the length of any side. To determine a square’s diagonal, this theorem is useful.

In the Triangle, A is the perpendicular, B is the base, and C is Hypotenuse.

Formula: Hypotenuse2 = Perpendicular2 + Base2

c2 = a2 + b2

**Conclusion**

Trigonometry may not have as many daily implementations but does make it easier to function with triangles. It is indeed a great addition to geometry and real values, and as such, it’s worth learning the fundamentals, although if users never want to go any further. Trigonometry is the branch of mathematics concerned with the relationship between the proportions of the sides of a right-angled triangle and their angles.

The trigonometric ratios used to study this relationship are sine, cosine, tangent, cotangent, secant, and cosecant. The fundamentals of trigonometry deal with angle measurement and angle problems. Trigonometry has three fundamental functions: sine, cosine, and tangent. Other important trigonometric functions can be derived from these three fundamental ratios or functional areas: cotangent, secant, and cosecant. These functions serve as the foundation for all of the essential topics covered in trigonometry. Trigonometric Ratios include six fundamental ratios that assist in building a relationship between both the proportion of sides of a triangle and the angle.

**FAQs**

**1. List the formulas of Trigonometry. **

Trigonometry Equations are accumulations of formulas that use mathematical identities to resolve issues encompassing the angles of the right-angle- triangle. These trigonometry equations involve complex numbers such as sine, cosine, secant, cosecant, tangent, and cosecant for given angles.

Pythagorean Identities

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

Sine and Cosine Law

- a/sinA = b/sinB = c/sinC
- c2 = a2 + b2 – 2ab cos C
- a2 = b2 + c2 – 2bc cos A
- b2 = a2 + c2 – 2ac cos B

**2. What is a trigonometric table?**

The trig table is simply a tabular collection of trig ratio values for various standard angles such as0°, 30°, 45°, 60°, and 90°, sometimes with other angles such as 180°, 270°, and 360° included. Because of patterns within trigonometric ratios and even between angles, it is simple to predict the trig table values and use the table as a reference to calculate trig values for various other angles. Trigonometric functions include sine, cosine, tan, cot, sec, and cosec.

**3. Define Trigonometric Ratios.**

There are six trigonometric ratios in trigonometry: sine, cosine, tangent, secant, cosecant, and cotangent.These ratios are abbreviated as sin, cos, tan, sec, cosec, and cot. Trigonometric ratios can calculate the ratios of two sides of a right triangle based on their respective angles. The values of these equivalent fractions can be determined by calculating using the acute angle measure in a right-angled triangle. An acute angle is less than 90° in length.

**4. Explain six Trigonometric Ratios.**

Sine: For any given angle, the sine ratio is defined as the ratio of the perpendicular to the hypotenuse. The sine of the angle in the given triangle can be written as sin = AB/AC.

- Cosine: For any given angle, the cosine ratio is defined as the ratio of the base to the hypotenuse. The cosine of the angle in the given triangle can be written as cos = BC/AC.
- Tangent: For any given angle, the tangent ratio is defined as the ratio of perpendicular to the base. The tangent of an angle in the given triangle can be written as tan = AB/BC.
- Cosecant: For any given angle, the cosecant ratio is defined as the ratio of the hypotenuse to the perpendicular. The cosecant of angle in the given triangle can be written as cosec = AC/AB.
- The secant ratio for any given angle is defined as the hypotenuse to base ratio. The secant of angle in the given triangle can be written as sec = AC/BC.
- Cotangent: The cotangent ratio is the base to the perpendicular ratio for any given angle. The cotangent of angle in the given triangle can be written as cot = BC/AB.

**5. Discuss the terms related to height and distance.**

Height and Distance include the line of sight is the line drawn between the eye of the observer and the point on the observed object. The horizontal level is the horizontal line seen through the observer’s eye. For objects above the horizontal level, the angle of elevation is important. It is the angle formed by the line of sight in relation to the horizontal level. For objects below the horizontal level, the angle of depression is important.

**6. List Important formulas of Trigonometry. **

Formulas related to Trigonometric Ratios are:

- sin(90° – A) = cos A
- cos(90° – A) = sin A
- tan(90° – A) = cot A
- cot(90° – A) = tan A
- sec(90° – A) = cosec A
- cosec(90° – A) = sec A
- sin2 θ + cos2 θ = 1 ⇒ sin2 θ = 1 – cos 2 θ ⇒ cos2θ = 1 – sin2 θ