Cot Half Angle Formula

Trigonometry word comes from a Greek word trigon means – triangle and metron mean – to measure. Initially, was concerned with missing parts of the triangle’s numerical values and its computing, if the value of other parts were given.

Trigonometric functions include many formulas. Apart from trigonometric identities and ratios, there are other formulas like half angle formulas.

Multiple Formulas for Cot Half Angle

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,

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.

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.

Examples of Cot half-angle formulas

Question 1: What is the value of Cot x/2 if cosec x = 2/1 and cot x = 3 / 4?

Solution:

If cosec x = 2/1 and cot x = 3 / 4

Then, Cot theta/2 =Cosec theta + cot theta

= 2/1 + ¾

=(8+3)/4

= 11/4

Question 2: What is the value of Cot x/2 if sin x = 3/4 and cos x = 4/5 ?

Solution:

If sin x = 3/4 and cos x = 4/5

Then, Cot theta/2 = Sin theta/(1 – Cos theta)

= ¾ +(1- ⅘)

=15/4

Question 3: If the value of cos A = 7/25, then find the value of cot A/2.

Solution:
Given,
cos A = 7/25

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)

What does the Cot Half Angle Formula mean?

The Cot Half Angle Formula cos A/2 = [(1 + cos A)/2]. Another cos formula in terms of semiperimeter is available. Cos (A/2) = [s (s – a)/bc] if a, b, and c are a triangle’s sides and A, B, and C are the triangle’s corresponding opposite angles.

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