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Cot 18 Degrees

The value of cot 18 degrees is 3.0776835. . .. Cot 18 degrees in radians is written as cot (18° × π/180°), i.e., cot (π/10) or cot (0.314159. . .). In this article, we will discuss the methods to find the value of cot 18 degrees with examples.

Cot 18°: √(5 + 2√5)
Cot 18° in decimal: 3.0776835. . .
Cot (-18 degrees): -3.0776835. . .
Cot 18° in radians: cot (π/10) or cot (0.3141592 . . .)

What is the Value of Cot 18 Degrees?

The value of cot 18 degrees in decimal is 3.077683537. . .. Cot 18 degrees can also be expressed using the equivalent of the given angle (18 degrees) in radians (0.31415 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 18 degrees = 18° × (π/180°) rad = π/10 or 0.3141 . . .
∴ cot 18° = cot(0.3141) = √(5 + 2√5) or 3.0776835. . .

Explanation:

For cot 18 degrees, the angle 18° lies between 0° and 90° (First Quadrant). Since cotangent function is positive in the first quadrant, thus cot 18° value = √(5 + 2√5) or 3.0776835. . .
Since the cotangent function is a periodic function, we can represent cot 18° as, cot 18 degrees = cot(18° + n × 180°), n ∈ Z.
⇒ cot 18° = cot 198° = cot 378°, and so on.
Note: Since, cotangent is an odd function, the value of cot(-18°) = -cot(18°).

Methods to Find Value of Cot 18 Degrees

The cotangent function is positive in the 1st quadrant. The value of cot 18° is given as 3.07768. . . We can find the value of cot 18 degrees by:

Using Trigonometric Functions
Using Unit Circle

Cot 18° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the cot 18 degrees as:

cos(18°)/sin(18°)
± cos 18°/√(1 - cos²(18°))
± √(1 - sin²(18°))/sin 18°
± 1/√(sec²(18°) - 1)
± √(cosec²(18°) - 1)
1/tan 18°

Note: Since 18° lies in the 1st Quadrant, the final value of cot 18° will be positive.

We can use trigonometric identities to represent cot 18° as,

tan (90° - 18°) = tan 72°
-tan (90° + 18°) = -tan 108°
-cot (180° - 18°) = -cot 162°

Cot 18 Degrees Using Unit Circle

To find the value of cot 18 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 18° angle with the positive x-axis.
The cot of 18 degrees equals the x-coordinate(0.9511) divided by y-coordinate(0.309) of the point of intersection (0.9511, 0.309) of unit circle and r.

Hence the value of cot 18° = x/y = 3.0777 (approx).

☛ Also Check:

Examples Using Cot 18 Degrees

Example 1: Find the value of 5 cot(18°)/9 cot(162°).

Solution:

Using trigonometric identities, we know, cot(18°) = -cot(180° - 18°) = -cot 162°.
⇒ cot(18°) = -cot(162°)
⇒ Value of 5 cot(18°)/9 cot(162°) = -5/9
Example 2: Using the value of cot 18°, solve: (cosec²(18°) - 1).

Solution:
We know, (cosec²(18°) - 1) = (cot²(18°)) = 9.4721
⇒ (cosec²(18°) - 1) = 9.4721
Example 3: Find the value of (cos (18°) cosec (9°) sec (9°))/2. [Hint: Use cot 18° = 3.0777]

Solution:

Using trigonometry formulas,
(cos (18°) cosec (9°) sec (9°))/2 = cos (18°)/(2 sin (9°) cos (9°))
Using sin 2a formula,
2 sin (9°) cos (9°) = sin (2 × 9°) = sin 18°
⇒ cos (18°) / sin (18°) = cot 18°
⇒ (cos (18°) cosec (9°) sec (9°))/2 = 3.0777

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FAQs on Cot 18 Degrees

What is Cot 18 Degrees?

Cot 18 degrees is the value of cotangent trigonometric function for an angle equal to 18 degrees. The value of cot 18° is √(5 + 2√5) or 3.0777 (approx).

What is the Value of Cot 18 Degrees in Terms of Cos 18°?

We know, using trig identities, we can write cot 18° as cos 18°/√(1 - cos²(18°)). Here, the value of cos 18° is equal to 0.951056.

How to Find the Value of Cot 18 Degrees?

The value of cot 18 degrees can be calculated by constructing an angle of 18° with the x-axis, and then finding the coordinates of the corresponding point (0.9511, 0.309) on the unit circle. The value of cot 18° is equal to the x-coordinate(0.9511) divided by the y-coordinate (0.309). ∴ cot 18° = 3.0777

What is the Value of Cot 18° in Terms of Sec 18°?

We can represent the cotangent function in terms of the secant function using trig identities, cot 18° can be written as 1/√(sec²(18°) - 1). Here, the value of sec 18° is equal to 1.0514.

How to Find Cot 18° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of cot 18° can be given in terms of other trigonometric functions as:

cos(18°)/sin(18°)
± cos 18°/√(1 - cos²(18°))
± √(1 - sin²(18°))/sin 18°
± 1/√(sec²(18°) - 1)
± √(cosec²(18°) - 1)
1/tan 18°

☛ Also check: trigonometric table

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