What are the Most Important Formulas for Trigonometry? & Identities

Most Important Formulas for Trigonometry and Identities

What is Trigonometry?

Trigonometry is the analysis of associations between various sections of circles and other geometrical figures that deal with angles, lengths, and heights of triangles and relationships. In engineering, astronomy, physics, and architectural design, applications of trigonometry are also described.

 

In mathematics, one of the most important subjects to study is trigonometry. Trigonometry is essentially a triangle analysis in which ‘Trigon’ means triangle and ‘metry’ means calculation.

 

Trigonometry, the mathematics branch, was concerned with unique angle functions and their application to calculations. There are six angle functions often used in trigonometry. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec) are their names and abbreviations (csc).

 

What are Trigonometric Identities?

Trigonometric identities are very helpful and studying the formulas helps to better solve the issues. There are a large number of areas where these trigonometry identities and trigonometry formula identities are used.

 

Some formulas that contain trigonometric functions are Trigonometric Identities. For all of the variables’ values, these trigonometry identities are valid. The relationship between calculating the angles and the length of the side of the right triangle is known as the trigonometric ratio.

 

Based on trigonometry ratios such as sine, cosine, tangent, cotangent, secant, and cosecant, the trigonometry formula list is given here. To address different trigonometry issues, these formulas are used.

 

What are the most important formulas for trigonometry?

For trigonometry, the most important formulas are those for the right triangle. If one of the angles in the triangle is an acute angle, then the proportion of the opposite side to the hypotenuse is the sin of the theta, the cosine is the proportion of the adjacent side to the hypotenuse, and the proportion of the opposite side to the adjacent side is the tangent.

 

We have Hypotenuse, Base, and Perpendicular in a right-angled triangle. The longest side is defined as hypotenuse, Perpendicular is the other side opposite the angle, and Base is the third side. Sine, cosine, secant, cosecant, tangent, cotangent, are the six trigonometric functions. Now all the trigonometric ratios are based on the lengths of the side of the triangles.

 

These calculations distinguish trigonometry from geometry, which discusses qualitative relationships primarily. This distinction, of course, is not necessarily absolute: the Pythagorean Theorem, for instance, is a statement in a right triangle about the lengths of the three sides and is thus quantitative in nature. Even, in its original form, trigonometry was a descendant of geometry by and large.

 

Applications of Trigonometry

In building, flight engineering, criminology, marine biology, engineering, and lots of other branches, trigonometry has enormous applications. The foundations of trigonometry are generally taught to students in high school (Class 9 or Class 10). Then, more nuanced topics discussed in class 11 and class 12 are added.

We will provide you with a full list of Trigonometry Formulas for Class 10, Trigonometry Formula Class 11, and Trigonometric Formulas for Class 12 to ensure you do not get confused with its components.

Reciprocal Relationship between Trigonometric Functions

The reciprocal relationship between different Trigonometric Functions are as under:

Tanθ =  1/cotθ⁡

Cosecθ  = 1/sinθ

secθ = 1/cos θ

 

Trigonometric Ratios of Complementary Angles

First Quadrant

sin(π/2−θθ) = cosθ

cos(π/2−θθ) = sinθ

tan(π/2−θθ) = cotθ

cot(π/2−θθ) = tanθ

sec(π/2−θθ) = cosecθ

cosec(π/2−θθ) = secθ

 

Second Quadrant

sin(π−θθ) = sinθ

cos(π−θθ) = -cosθ

tan(π−θθ) = -tanθ

cot(π−θθ) = -cotθ

sec(π−θθ) = -secθ

cosec(π−θθ) = cosecθ

 

Third Quadrant

sin(π+θθ) = -sinθ

cos(π+θθ) = -cosθ

tan(π+θθ) = tanθ

cot(π+θθ) = cotθ

sec(π+θθ) = -secθ

cosec(π+θθ) = -cosecθ

 

Fourth Quadrant

sin(2π−θθ) = -sinθ

cos(2π−θθ) = cosθ

tan(2π−θθ) = -tanθ

cot(2π−θθ) = -cotθ

sec(2π−θθ) = secθ

cosec(2π−θθ) = -cosecθ

 

Periodicity Identities

sin(2nπ + θθ) = sinθ

cos(2nπ + θθ) = cosθ

tan(2nπ + θθ) = tanθ

cot(2nπ + θθ) = cotθ

sec(2nπ + θθ) = secθ

cosec(2nπ + θθ) = cosecθ

 

Trigonometry Table

Trigonometry table is a table that you can refer to for the values of trigonometric ratios 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

 

Download PDF List of trigonometry formulas

List of Important Essentials of Trigonometry 

Trigonometry is a triangle-dealing branch of mathematics. Trigonometry connects the triangle lengths and angles. Trigonometry and its formulas have a large number of applications. In geography, for example, the triangulation method is used to calculate the distance between landmarks; in astronomy, the distance to neighbouring stars is calculated.

 

Various kinds of problems can be solved using trigonometry formulas. Trigonometric ratios (sin, cos, tan, sec, cosec, and tan), Pythagorean identities, product identities, etc. are included in these problems. There are some more formulas, including the sign of ratios in various quadrants which include identities of co-function (shifting angles), identities of sum & difference, identities of double angles.

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