Cot Half Angle Formula

Trigonometry is a branch of mathematics that uses trigonometric ratios to determine the angles and incomplete sides of a triangle. The trigonometric ratios such as sine, cosine, tangent, cotangent, secant, and cosecant are used to investigate this branch of mathematics. It’s the study of how the sides and angles of a right-angled triangle are related.

In this article, we have covered formulas related to the cot half angle formula, its derivation-related examples and others in detail.

Cotangent ratio is expressed as the ratio of the length of the adjacent side of an angle divided by the length of the opposite side. It is denoted by the symbol cot.

If θ is the angle that lies between the base and hypotenuse of a right-angled triangle then,

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

In trigonometry, half-angle formulas are usually represented as θ/2, where θ is the angle. The half-angle equations are used to determine the precise values of trigonometric ratios of standard angles such as 30°, 45°, and 60°. We may get the ratio values for complex angles like 22.5° (half of 45°) or 15° (half of 30°) by using the ratio values for these ordinary angles.

Cotangent half-angle is denoted by the abbreviation cot θ/2. It’s a trigonometric function that returns the cot function value for half angle. The period of the function cot θ is π, but the period of cot θ/2 is 2π.

Formula for cot(θ/2) is represented below:

Derivation of Cot Half Angle Formula

Formula for cotangent half angle is derived by using the half angle formula for sine and cosine.

We know that,

• sin θ/2 = ±√((1 – cos θ) / 2)…(i)

cos θ/2 = √(1 – (√((1 – cos θ) / 2))²)

cos θ/2 = √(1 – ((1 – cos θ)/ 2))

cos θ/2 = √((2 – 1 + cos θ)/ 2)

• cos θ/2 = ±√((1 + cos θ)/ 2)…(ii)

Also, cot θ/2 = cos (θ/2)/ sin (θ/2)

So,

cot θ/2 = √((1 + cos θ)/ 2)/√((1 – cos θ)/ 2)

cot θ/2 = √((1 + cos θ)/(1 – cos θ))

Thus, formula for cotangent half angle ratio is derived.

Read More:

• Sine Half Angle Formula
• Double Angle Formulas
• Trigonometry Formulas

Example 1. If cos θ = 3/5, find the value of cot θ/2 using the half-angle formula.

Solution:

We have, cos θ = 3/5

Using the formula we get,

cot θ/2 = √((1 + cos θ)/(1 – cos θ))

= √((1 + (3/5))/ (1 – (3/5)))

= √((8/5)/ (2/5))

= √4 = ±2

Example 2. If cos θ = 12/13, find the value of cot θ/2 using the half-angle formula.

Solution:

We have, cos θ = 12/13

Using the formula we get,

cot θ/2 = √((1 + cos θ)/(1 – cos θ))

= √((1 + (12/13))/ (1 – (12/13)))

= √((25/13)/ (1/13))

= √25 = ±5

Example 3. If sin θ = 8/17, find the value of cot θ/2 using the half-angle formula.

Solution:

We have, sin θ = 8/17

Find the value of cos θ using the formula sin² θ + cos² θ = 1.

cos θ = √(1 – (64/289))

= √(225/289)

= 15/17

Using the formula we get,

cot θ/2 = √((1 + cos θ)/(1 – cos θ))

= √((1 + (15/17))/ (1 – (15/17)))

= √((32/17)/ (2/17))

= √16 = ±4

Example 4. If sec θ = 5/4, find the value of cot θ/2 using the half-angle formula.

Solution:

We have, sec θ = 5/4

Using cos θ = 1/sec θ, we get cos θ = 4/5.

Using the formula we get,

cot θ/2 = √((1 + cos θ)/(1 – cos θ))

= √((1 + (4/5))/ (1 – (4/5)))

= √((9/5)/ (1/5))

= √9 = ±3

Example 5. If tan θ = 12/5, find the value of cot θ/2 using the half-angle formula.

Solution:

We have, tan θ = 12/5

Clearly, cos θ = 5/√(12² + 5²) = 5/13

Using the formula we get,

cot θ/2 = √((1 + cos θ)/(1 – cos θ))

= √((1 + (5/13))/ (1 – (5/13)))

= √((18/13)/ (8/5))

= √(18/8)

= √(9/4) = ±3/2

Example 6. If cot θ = 8/15, find the value of cot θ/2 using the half-angle formula.

Solution:

We have, cot θ = 8/15

Clearly, cos θ = 8/√(8² + 15²) = 8/17

Using the formula we get,

cot θ/2 = √((1 + cos θ)/(1 – cos θ))

= √((1 + (8/17))/ (1 – (8/17)))

= √((25/17)/ (9/17))

= √(25/9) = ±5/3

Example 7. Find the value of cot 15° using the half-angle formula.

Solution:

We have to find the value of cot 15°

Let us take θ/2 = 15°

=> θ = 30°

Using the half angle formula we have,

cot θ/2 = √((1 + cos θ)/(1 – cos θ))

= √((1 + cos 30°)/ (1 – cos 30°))

= √((1 + (√3/2))/ (1 – (√3/2)))

= √((2 + √3)/ (2 – √3))

= √(((2 + √3) (2 + √3))/ ((2 – √3) (2 + √3)))

= √((4 + 3 + 4√3)/ (4 – 3))

= ±√(7 + 4√3)

What is the half angle formula?

Half angle formulas are formulas used to solve trigonometric function when the given angle is half of of any known angle. For example we will find cot (θ/2) using the value of cos θ, and so on.

What is the formula for cot?

Cot of any angle is found using the formula, cot θ = Base /Perpendicular.

What is the double angle formula for cot?

Double angle formula for cot is, cot 2θ = (cot²θ – 1)/(2cotθ)

What does the Half Angle Sin Formula mean?

Half sine angle formula gives the value of sin (θ/2), i.e. sin (θ/2) = [(1 – cos θ) / 2]