Trigonometry Formulas & Identities (Complete List)

Trigonometry Formulas & Identities

What is a trigonometric formula?

A trigonometric Formula is used to calculate the area, volume, and circumference of any shape or object. It is also used to determine the length, breadth, height, and diameter of an area, for example, “1 / 2 * 3/ 4 inches”. In practice, trigonometry is mostly used for measuring volume but can be used for calculating area. A trigonometric formula for length is: l = 1 / 16 / ((2 / 0.75) = 1/16, where 0.75 is the ratio between an area and its length.

What are all types of trigonometric calculations?

A table contains a list of trigonometric calculations. The first type of calculation (as defined by Pythagoras) is the common-size triangle. This formula is given above for its length and area. It is also used for calculating area, as well as the other three, in various formulas.

The second type of trigonometric formula is known as the right-angle triangular area. Below you can see how it differs from the left-angle triangle. For any size right-angle triangles or any even right-angled rectangle, this is the one and only type of trigonometric calculation, as it has no perimeter. Calculating area only requires that we have the following equation or expression: 1 / (l x b + d )/ 2 or the equivalent equivalent expression using an imaginary number system: Area = Length / Width

Where:

Area = Length / Width

B = Depth / Width

d = (Diameter)/(Depth/2)

L = Measurement of length

a = Angle of incidence of point on triangle

b = Angle of maximum angle to base

The next type of trigonometric formula that only comes from Euclidean geometry is the trapezoid area. A trapezoid area is simply a plane that passes through the sides that form its outer wall to one side of its interior wall. Like a square, a trapezoid area can contain only four sides that make up the length of the entire plane, regardless of whether the plane passing through the sides is vertical or horizontal.

The third type of trigonometric formula is an area that is greater than a whole plane, or less than a whole plane, but greater than half the plane or less is still better than nothing. Such areas are called cubes because if two cubes were put together they would all fit into the same space. In such cases, the area would be divided equally among each smaller cube, while remaining intact and unbroken enough to permit them to fit in a larger space.

Is there a difference between a cubic and a quadratic equation?

A cricoid and a square root are the primary differences between these types of equations. When looking at the equation describing the cricoid, imagine that a large circle is divided into sections. Let’s say that c is the longest section of the ring and q is the shortest section of the ring. It turns out that when the short section of the ring is divided in half, the long section falls into the bottom of the short section, meaning that c is longer than q. This is also true with squares and numbers. If the top part of the square is shorter than the bottom part, then the square has a closer distance than the number 5. If the top half of the square is shorter than the bottom half, then the square has a farther distance than the square root 2. Because there is more of a distance between the top and bottom, the top part becomes bigger.

A cricoid and the square root are also different when viewed from another angle. First, the equation describes the cricoid that measures out the lengths of the segments using only triangles. Second, the definition describes only those segments that meet the plane, not those that extend from it. Finally, the cricoid in itself does not have a perimeter, which means that you cannot measure the length of any segment inside an edge of the cricoid. Therefore, when measuring the area of cricoid, the area of cricoid is the sum of the lengths of those segments. As you can see, the trapezoid area doesn’t have that perimeter problem, so a cricoid is the best option to use when trying to find that area.

Trigonometry Formulas List

When we learn about trigonometric formulas, we consider them for right-angled triangles only. In a right-angled triangle, we have 3 sides namely – Hypotenuse, Opposite side (Perpendicular), and Adjacent side (Base). The longest side is known as the hypotenuse, the side opposite to the angle is perpendicular and the side where both hypotenuse and opposite side rests is the adjacent side.

Also Read:- Multiplication Table from 2 to 20

Basic Trigonometric Function Formulas

There are basically 6 ratios used for finding the elements in Trigonometry. They are called trigonometric functions. The six trigonometric functions are sine, cosine, secant, co-secant, tangent, and co-tangent.

By using a right-angled triangle as a reference, the trigonometric functions and identities are derived:

  • sin θ = Opposite Side/Hypotenuse
  • cos θ = Adjacent Side/Hypotenuse
  • tan θ = Opposite Side/Adjacent Side
  • sec θ = Hypotenuse/Adjacent Side
  • cosec θ = Hypotenuse/Opposite Side
  • cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

The Reciprocal Identities are given as:

  • cosec θ = 1/sin θ
  • sec θ = 1/cos θ
  • cot θ = 1/tan θ
  • sin θ = 1/cosec θ
  • cos θ = 1/sec θ
  • tan θ = 1/cot θ

All these are taken from a right-angled triangle. When the height and base side of the right triangle are known, we can find out the sine, cosine, tangent, secant, cosecant, and cotangent values using trigonometric formulas. The reciprocal trigonometric identities are also derived by using the trigonometric functions.

Trigonometry Table

Below is the table for trigonometry formulas for angles that are commonly used for solving problems.

Angles (In Degrees) 30° 45° 60° 90° 180° 270° 360°
Angles (In Radians) π/6 π/4 π/3 π/2 π 3π/2
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
csc 2 √2 2/√3 1 -1
sec 1 2/√3 √2 2 -1 1

Periodicity Identities (in Radians)

These formulas are used to shift the angles by π/2, π, 2π, etc. They are also called co-function identities.

  • sin (π/2 – A) = cos A & cos (π/2 – A) = sin A
  • sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A
  • sin (3π/2 – A)  = – cos A & cos (3π/2 – A)  = – sin A
  • sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A
  • sin (π – A) = sin A &  cos (π – A) = – cos A
  • sin (π + A) = – sin A & cos (π + A) = – cos A
  • sin (2π – A) = – sin A & cos (2π – A) = cos A
  • sin (2π + A) = sin A & cos (2π + A) = cos A

All trigonometric identities are cyclic in nature. They repeat themselves after this periodicity constant. This periodicity constant is different for different trigonometric identities. tan 45° = tan 225° but this is true for cos 45° and cos 225°. Refer to the above trigonometry table to verify the values.

The three fundamental identities are:

1. sin2 A + cos2 A = 1
2. 1+tan2 A = sec2 A
3. 1+cot2 A = csc2 A

Co-function Identities (in Degrees)

The co-function or periodic identities can also be represented in degrees as:

  • sin(90°−x) = cos x
  • cos(90°−x) = sin x
  • tan(90°−x) = cot x
  • cot(90°−x) = tan x
  • sec(90°−x) = csc x
  • csc(90°−x) = sec x

Sum & Difference Identities

  • sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
  • cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
  • tan(x+y) = (tan x + tan y)/ (1−tan x •tan y)
  • sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
  • cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
  • tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)

Double Angle Identities

  • sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)]
  • cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)]
  • cos(2x) = 2cos2(x)−1 = 1–2sin2(x)
  • tan(2x) = [2tan(x)]/ [1−tan2(x)]
  • sec (2x) = secx/(2-sec2 x)
  • csc (2x) = (sec x. csc x)/2


Triple Angle Identities

  • Sin 3x = 3sin x – 4sin3x
  • Cos 3x = 4cos3x-3cos x
  • Tan 3x = [3tanx-tan3x]/[1-3tan2x]

Inverse Trigonometry Formulas

  • sin-1 (–x) = – sin-1 x
  • cos-1 (–x) = π – cos-1 x
  • tan-1 (–x) = – tan-1 x
  • cosec-1 (–x) = – cosec-1 x
  • sec-1 (–x) = π – sec-1 x
  • cot-1 (–x) = π – cot-1 x

What is Sin 3x Formula?

Sin 3x is the sine of three times of an angle in a right-angled triangle, that is expressed as:

Sin 3x = 3sin x – 4sin3x

Trigonometry Formulas From Class 10 to Class 12

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