Cot Half Angle Formula

Cot Half Angle Formula

A branch of Mathematics known as Trigonometry uses trigonometric ratios to calculate the angles and incomplete sides of triangles. This area of Mathematics is studied using trigonometric ratios like sine, cosine, tangent, cotangent, secant, and cosecant. It is the study of the relationships between a right-angled triangle’s sides and angles. The words “Trigonon” and “Metron,” which stand for a triangle and a measurement, respectively, make up the term “Trigonometry.” This relationship enables the estimation of a right-angled triangle’s unknown dimensions through the use of equations and identities.

A trigonometric ratio is the ratio of any two sides of a right triangle’s lengths. These ratios connect the ratio of a right triangle’s sides to the angle in trigonometry. The adjacent side of an angle’s length divided by the opposite side’s length is used to express the Cot Half Angle Formula ratio. The phrase Cot Half Angle Formula is used to represent it.

If θ the angle between a right-angled triangle’s base and the hypotenuse is, then 

cot θ = Base/Perpendicular = cos θ/ sin θ.

In this case, the angle’s base side is next to it, and the angle’s perpendicular side is across from it.

Half-angle formulas in Trigonometry are typically written as θ/2, where θ is the angle. To calculate the precise values of trigonometric ratios of common angles like 30°, 45°, and 60°, half-angle equations are utilised. Using the ratio values for these regular angles, students can obtain the ratio values for complex angles like 22.5° (half of 45°) or 15° (half of 30°). The symbol cot is used to indicate the Cot Half Angle Formula. The trigonometric function’s return value is the value of the Cot Half Angle Formula function. While the period of the function Cot Half Angle Formula is, cot/2 has a period of 2.

An Introduction to Trigonometry Half-Angle Formulas

In trigonometry, students learn Cot Half Angle Formula (also known as half-angle identities). The double-angle formulas can be used to derive Cot Half Angle Formula. Students are aware that the angle sum and difference formulae in trigonometry can be used to obtain the double angle formulas. Cot Half Angle Formula typically use the notation x/2, A/2, /2, etc. to indicate half-angles, which are sub-multiple angles. The trigonometric ratios of angles like 22.5° (half of the standard angle 45°), 15° (half of the standard angle 30°), etc. are precisely calculated using the half angle formulas.

On the Extramarks website, students could peruse the Cot Half Angle Formula, their justifications, and several comprehensively solved examples.

The sin, cos, and tan half-angle formulas are shown in this section. The trigonometric table provides the values of the trigonometric functions (sin, cos, tan, cot, sec, cosec) for angles such as 0°, 30°, 45°, 60°, and 90°. However, the half-angle formulas are very helpful for knowing the precise values of sin 22.5°, tan 15°, and others. Additionally, they aid in the proof of a number of trigonometric identities. From the double-angle formulae, students may deduce half-angle formulas, which are represented in terms of half angles like /2, x/2, A/2, etc. The list of significant half-angle formulas is as follows: 

Half angle formula of sin: sin A/2 = ±√[(1 – cos A) / 2]

Half angle formula of cos: cos A/2 = ±√[(1 + cos A) / 2]

Half angle formula of tan: tan A/2 = ±√[1 – cos A] / [1 + cos A] (or) sin A / (1 + cos A) (or) (1 – cos A) / sin A

Half-Angle Formula

Students must first deduce the following half-angle formulas in order to obtain the aforementioned formulas. Double angles like 2, 2A, 2x, etc. are used in the double angle formulas. The double-angle formulas for sin, cos, and tan are known to be

sin 2x = 2 sin x cos x

cos 2x = cos2 x – sin2 x (or)

= 1 – 2 sin2x (or)

= 2 cos2x – 1

tan 2x = 2 tan x / (1 – tan2x)

Cot Half Angle Formula identities are obtained if students swap out x for A/2 on both sides of every equation involving a double-angle formula (2x = 2(A/2) = A).

sin A = 2 sin(A/2) cos(A/2)

cos A = cos2 (A/2) – sin2 (A/2) (or)

= 1 – 2 sin2 (A/2) (or)

= 2 cos2(A/2) – 1

tan A = 2 tan (A/2) / (1 – tan2(A/2)

Double‐Angle and Half‐Angle Identities

Students can also create their own Cot Half Angle Formula by combining two existing ones. For instance, students may deduce three crucial Cot Half Angle Formula identities for sin, cos, and tan, which are mentioned in the first section, solely from the formula for cos A. Here is the proof for the Cot Half Angle Formula.

The double-angle identities and the Cot Half Angle Formula identities are special cases of the sine and cosine sum and difference formulas. First, using the sine’s sum identity,

sin 2α = sin (α + α)

sin 2α = sin α cos α + cos α sin α

sin 2α = 2 sin α cos α

Trigonometry Angles and their Notations

The need to calculate angles and distances in disciplines like Astronomy, Mapmaking, Surveying, and Artillery Range finding led to the development of Trigonometry. Plane Trigonometry deals with issues involving angles and lengths in a single plane. Spherical Trigonometry takes into account applications to similar issues in more than one plane of three-dimensional space.

Multiple Formulas for Cot Half Angle

The name “Trigonometry” is derived from the Greek words “trigon” (triangle) and “meton” (to measure). At first, students were concerned about the computation of the triangle’s missing numerical values if the values of the other portions were known. Formulas for trigonometric functions are numerous. There are other formulas besides trigonometric identities and ratios, such as half-angle formulas.

Cotangent Trigonometric Ratio

The Cot Half Angle Formula is the tangent’s reciprocal in Trigonometry. It is the tangent’s reciprocal or multiplicative inverse, using the formula tan cot = 1. The ratio of the adjacent leg (b) to the opposite leg is known as the Cot Half Angle Formula in a right triangle.

Cot Half Angle (Cot θ/2) Formula

Finding the values of the trigonometric functions for half-angles might be quite important at times. For instance, students can change an equation with exponents into one without exponents and whose angles are multiples of the original angle by applying some half-angle formula. It should be noticed that from double-angle formulas, students obtain half-angle formulas. The double-angle formula for the cosine yields both sin (2A) and cos (2A).

The following is the formula for half-angle identities:

Half angle for Cotangent Cosec theta + cot theta

Half angle for Cotangent (1 + cos theta) /sin theta

Half angle for Cotangent ±Sqrt (1+cos theta/1 – cos theta)

Half angle for Cotangent Sin theta/(1 – Cos theta)

Examples of Cot half-angle formulas

Students will now demonstrate how to use the sine function’s half-angle formula. With one of the cos A formulae mentioned above,  here is,

cos A = 1 – 2 sin2 (A/2)

From this,

2 sin2 (A/2) = 1 – cos A

sin2 (A/2) = (1 – cos A) / 2

sin (A/2) = ±√[(1 – cos A) / 2]