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. Trigonometry is made up of the words ‘Trigonon’ and ‘Metron,’ which represent a triangle and a measurement, respectively. By applying equations and identities based on this connection, it facilitates the estimation of unknown dimensions of a right-angled triangle. 

Cotangent Trigonometric Ratio

The ratio of the lengths of any two sides of a right triangle is called a trigonometric ratio. In trigonometry, these ratios link the ratio of sides of a right triangle to the angle. The 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

Here, base is the side adjacent to the angle and perpendicular is the side opposite to it.

Cot Half Angle (Cot θ/2) Formula

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. The 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.

cot θ/2 = √((1 + cos θ)/(1 – cos θ))

Derivation

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

We know, sin θ/2 = ±√((1 – cos θ) / 2).

Find cos θ/2 using the identity sin² θ + cos² θ = 1.

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)

Also, we know cot θ/2 = cos (θ/2)/ sin (θ/2).

So, we get

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

cot θ/2 = √((1 + cos θ)/(1 – cos θ))

This derives the formula for cotangent half angle ratio.

Sample Problems

Problem 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

Problem 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

Problem 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

Problem 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

Problem 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

Problem 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

Problem 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)