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Cot 120 Degrees

The value of cot 120 degrees is -0.5773502. . .. Cot 120 degrees in radians is written as cot (120° × π/180°), i.e., cot (2π/3) or cot (2.094395. . .). In this article, we will discuss the methods to find the value of cot 120 degrees with examples.

Cot 120°: -(1/√3)
Cot 120° in decimal: -0.5773502. . .
Cot (-120 degrees): 0.5773502. . . or 1/√3
Cot 120° in radians: cot (2π/3) or cot (2.0943951 . . .)

What is the Value of Cot 120 Degrees?

The value of cot 120 degrees in decimal is -0.577350269. . .. Cot 120 degrees can also be expressed using the equivalent of the given angle (120 degrees) in radians (2.09439 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 120 degrees = 120° × (π/180°) rad = 2π/3 or 2.0943 . . .
∴ cot 120° = cot(2.0943) = -(1/√3) or -0.5773502. . .

Explanation:

For cot 120 degrees, the angle 120° lies between 90° and 180° (Second Quadrant). Since cotangent function is negative in the second quadrant, thus cot 120° value = -(1/√3) or -0.5773502. . .
Since the cotangent function is a periodic function, we can represent cot 120° as, cot 120 degrees = cot(120° + n × 180°), n ∈ Z.
⇒ cot 120° = cot 300° = cot 480°, and so on.
Note: Since, cotangent is an odd function, the value of cot(-120°) = -cot(120°).

Methods to Find Value of Cot 120 Degrees

The cotangent function is negative in the 2nd quadrant. The value of cot 120° is given as -0.57735. . . We can find the value of cot 120 degrees by:

Using Unit Circle
Using Trigonometric Functions

Cot 120 Degrees Using Unit Circle

To find the value of cot 120 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 120° angle with the positive x-axis.
The cot of 120 degrees equals the x-coordinate(-0.5) divided by y-coordinate(0.866) of the point of intersection (-0.5, 0.866) of unit circle and r.

Hence the value of cot 120° = x/y = -0.5774 (approx).

Cot 120° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the cot 120 degrees as:

cos(120°)/sin(120°)
± cos 120°/√(1 - cos²(120°))
± √(1 - sin²(120°))/sin 120°
± 1/√(sec²(120°) - 1)
± √(cosec²(120°) - 1)
1/tan 120°

Note: Since 120° lies in the 2nd Quadrant, the final value of cot 120° will be negative.

We can use trigonometric identities to represent cot 120° as,

tan (90° - 120°) = tan(-30°)
-tan (90° + 120°) = -tan 210°
-cot (180° - 120°) = -cot 60°

☛ Also Check:

Examples Using Cot 120 Degrees

Example 1: Find the value of 8 cot(120°)/9 cot(60°).

Solution:

Using trigonometric identities, we know, cot(120°) = -cot(180° - 120°) = -cot 60°.
⇒ cot(120°) = -cot(60°)
⇒ Value of 8 cot(120°)/9 cot(60°) = -8/9
Example 2: Find the value of (cos (120°) cosec (60°) sec (60°))/2. [Hint: Use cot 120° = -0.5774]

Solution:

Using trigonometry formulas,
(cos (120°) cosec (60°) sec (60°))/2 = cos (120°)/(2 sin (60°) cos (60°))
Using sin 2a formula,
2 sin (60°) cos (60°) = sin (2 × 60°) = sin 120°
⇒ cos (120°) / sin (120°) = cot 120°
⇒ (cos (120°) cosec (60°) sec (60°))/2 = -0.5774
Example 3: Simplify: 5 (cot 120°/tan(-30°))

Solution:

We know cot 120° = tan(-30°)
⇒ 5 cot 120°/tan(-30°) = 5 (cot 120°/cot 120°)
= 5(1) = 5

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FAQs on Cot 120 Degrees

What is Cot 120 Degrees?

Cot 120 degrees is the value of cotangent trigonometric function for an angle equal to 120 degrees. The value of cot 120° is -(1/√3) or -0.5774 (approx).

How to Find the Value of Cot 120 Degrees?

The value of cot 120 degrees can be calculated by constructing an angle of 120° with the x-axis, and then finding the coordinates of the corresponding point (-0.5, 0.866) on the unit circle. The value of cot 120° is equal to the x-coordinate(-0.5) divided by the y-coordinate (0.866). ∴ cot 120° = -0.5774

How to Find Cot 120° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of cot 120° can be given in terms of other trigonometric functions as:

cos(120°)/sin(120°)
± cos 120°/√(1 - cos²(120°))
± √(1 - sin²(120°))/sin 120°
± 1/√(sec²(120°) - 1)
± √(cosec²(120°) - 1)
1/tan 120°

☛ Also check: trigonometric table

What is the Exact Value of Cot 120 Degrees?

The exact value of cot 120 degrees can be given accurately up to 8 decimal places as -0.57735026 or as -(1/√3).

What is the Value of Cot 120 Degrees in Terms of Tan 120°?

Since the cotangent function is the reciprocal of the tangent function, we can write cot 120° as 1/tan(120°). The value of tan 120° is equal to -√3.

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