Half-Angle Formula: Definition and Explanation

Mathematics is a domain in which trigonometry is one of the most important branches. According to Hipparchus, this concept originated in Greece. The purpose of this topic is to explore the topic of trigonometric ratios and the formula of the half-angle value with examples. So, let's get started!

Keyterms: Trigonometry, Trigonometric ratios, Half-angle value, Angles, Triangles, Right-angled triangles, Radians, Trigonometry angles, Adjacent sides

Read More: Trigonometry Table

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The concept of trigonometry 

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The study of trigonometry involves angles and sides of triangles as well as the angles themselves. In particular, it involves right-angled triangles. The trigonometric ratios are used to find the angles and sides of a triangle. The trigonometric ratios are one of the elements of mathematics. Different formulas are available for calculating the triangle as well as the half-angle.

Depending on the angle, right-angled triangles are measured either in radians or degrees. There are five common trigonometry angles: 0°, 30°, 45°, 60°, and 90°. These angles can be easily used with various formulas to display sides. In addition, there are formulas for half-angle values, which are also widely used.

Plane trigonometry and spherical geometry are two sub-branches of this branch of mathematics. In trigonometric functions, triangles are represented as trigonometric ratios. The sine, cosine, and tangent functions are integral to trigonometry. Let's consider a right-angled triangle whose hypotenuse is its longest side. The adjacent and opposite sides of the hypotenuse are referred to as opposing and adjacent sides.

Listed below are some important half-angle formulas:

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Identities at a half-angle

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The most commonly used half-angle identities in solving trigonometry problems are as follows:

• The formula for half angles of sin is: sin A/2 = ±√[(1 - cos A) / 2]
• In half angles, cos A/2 = ±√[(1 + cos A) / 2]
• The half-angle formula for tan is: tan A/2 = ±√[1 - cos A] / [1 + cos A] (or) sin A / (1 + cos A) (or) (1 - cos A) / sin A

The half-angle formulas are derived from the double angle formulas

Here are the formulas for calculating half angles, which we can use to derive the above formulas. There are a wide variety of double angle formulas in the form of 2θ, 2A, 2x, etc. It is well known that the double angle formulas for sin, cos, and tan are

• 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)

Every equation of a double angle formula may be solved by substituting A/2 for x on both sides (since 2x = 2A/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))

You can also obtain one half-angle formula by using another half-angle formula. For example, we can deduce three important half-angle identities from the formula of cos A in the first section. See below for the proof of half-angle formulas.

Read More: Isosceles Triangle Theorems

The Half Angle Formula for Proof of Sin

The next step will be to prove the half-angle formula for the sine function. Let's use one of the formulas of cos A.

Cos A = 1 - 2 sin² (A/2)

As a result,

2 Sin² (A/2) = 1 - cos A

Sin² (A/2) = (1 - cos A) / 2

Sin (A/2) = ±√[(1 - cos A) / 2]

A Half Angle Formula for Cos Derivation

Our next step is to prove the half-angle formula for the cosine function. Let us choose one of the above formulas for cos A.

Cos A = 2 cos²(A/2) - 1

As a result,

2 Cos²(A/2) = 1 + cos A

Cos² (A/2) = (1 + cos A) / 2

Cos (A/2) = ±√[(1 + cos A) / 2]

Formula for the Tan Derivation at Half Angles

The formula for tan 

(A/2) is [sin (A/2)] / [cos (A/2)]

Using the half angle formulas of sin and cos,

tan (A/2) 

= [±√(1 - cos A)/2] / [±√(1 + cos A)/2] 

= ±√[(1 - cos A) / (1 + cos A)]

It is one of the formulas for tan (A/2). Here, let us rationalize the denominator and derive the other two formulas.

tan (A/2) 

= ±√[(1 - cos A) / (1 + cos A)] × √[(1 - cos A) / (1 - cos A)] 

= √[(1 - cos A)² / (1 - cos2A)]

= √[(1 - cos A)²/ sin2A] 

= (1 - cos A) / sin A

The second formula of tan (A/2) can be found here. We can derive another formula by multiplying and dividing the above formula by (1 + cos A). This gives us

tan (A/2) 

= [(1 - cos A) / sin A] × [(1 + cos A) / (1 + cos A)] 

= (1 - cos²A) / [sin A (1 + cos A)]

= sin²A / [sin A (1 + cos A)] 

= sin A / (1 + cos A)

Thus, tan (A/2) = ±√[(1 - cos A) / (1 + cos A)] = (1 - cos A) / sin A = sin A / (1 + cos A).

Formula for half angles using semi-perimeters

Our next section will show you how to calculate the half-angle formula using the semi perimeter. In other words, these are the half-angle formulas in terms of triangle sides. Here's an example of a triangle ABC, where AB = c, BC = a, and CA = b.

Formula for half angles using semi perimeters

This is one of the formulas we can use. Triangles have semi-perimeters 

s = (a + b + c)/2.

In this case, 

2s = a + b + c. 

According to one of the formulas above,

Cos A = 2 cos²(A/2) - 1 

(or) 2 Cos²(A/2) = 1 + cos A

Applying the law of cosines,

2 Cos²(A/2) = 1 + [(b² + c² - a²) / (2bc)]

2 Cos²(A/2) = [2bc + b² + c² - a²] / [2bc]

2 Cos²(A/2) = [ (b + c)² - a²] / [2bc]………. [Using (a+b)² formula]

2 Cos²(A/2) = [ (b + c + a) (b + c - a) ] / [2bc]………. [Using a² - b² formula]

2 Cos²(A/2) = [2s (2s - 2a)] / [2bc]………. [As 2s = a + b + c]

2 Cos²(A/2) = [ 2s (s - a) ] / [bc]

Cos²(A/2) = [s(s - a) ] / [bc]

Cos (A/2) = √[ s (s - a) ] / [bc]

A half-angle formula has been derived for the cosine of angle A/2. In the same way, we can deduce the half-angle identities of cosine by means of the semi perimeter. The semi perimeter can also be used to deduce the half-angle identities of sine.

Sin²(A/2) 

= (1 − cos A)/2 

= (1/2)[1−(b²+c²−a²)/2bc]………. (Using the law of cosines)

= (1/2)(a²−(b−c)²)/2bc 

= (1/2)(a + b − c)(a + c − b)/2bc 

= (1/2){(a + b + c) − 2c}{(a + b + c) − 2b}/2bc 

= (1/2)(2s − 2c)(2s − 2b)/2bc 

= (s − b)(s − c)/bc

⇒ sin (A/2) = √[(s − b)(s − c)/bc]

In a similar manner, we can derive other half-angle sine function formulas. The formula tan(A/2) = sin(A/2)/cos(A/2) can be used to calculate half-angle tangent functions.

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Things to Remember

• The half-angle formulas are derived from the double angle formulas.
• The formula for half angles of sin is: sin A/2 = ±√[(1 - cos A) / 2]
• In half angles, cos A/2 = ±√[(1 + cos A) / 2]
• The half-angle formula for tan is: tan A/2 = ±√[1 - cos A] / [1 + cos A] (or) sin A / (1 + cos A) (or) (1 - cos A) / sin A
• Cos (A/2) = √[ s (s - a) ] / [bc]
• sin (A/2) = √[(s − b)(s − c)/bc]

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