Tan 2pi/3

The value of tan 2pi/3 is -1.7320508. . .. Tan 2pi/3 radians in degrees is written as tan ((2π/3) × 180°/π), i.e., tan (120°). In this article, we will discuss the methods to find the value of tan 2pi/3 with examples.

• Tan 2pi/3: -√3
• Tan 2pi/3 in decimal: -1.7320508. . .
• Tan (-2pi/3): 1.7320508. . . or √3
• Tan 2pi/3 in degrees: tan (120°)

What is the Value of Tan 2pi/3?

The value of tan 2pi/3 in decimal is -1.732050807. . .. Tan 2pi/3 can also be expressed using the equivalent of the given angle (2pi/3) in degrees (120°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 2pi/3 radians = 2pi/3 × (180°/pi) = 120° or 120 degrees
∴ tan 2pi/3 = tan 2π/3 = tan(120°) = -√3 or -1.7320508. . .

Explanation:

For tan 2pi/3, the angle 2pi/3 lies between pi/2 and pi (Second Quadrant). Since tangent function is negative in the second quadrant, thus tan 2pi/3 value = -√3 or -1.7320508. . .
Since the tangent function is a periodic function, we can represent tan 2pi/3 as, tan 2pi/3 = tan(2pi/3 + n × pi), n ∈ Z.
⇒ tan 2pi/3 = tan 5pi/3 = tan 8pi/3 , and so on.
Note: Since, tangent is an odd function, the value of tan(-2pi/3) = -tan(2pi/3).

Methods to Find Value of Tan 2pi/3

The tangent function is negative in the 2nd quadrant. The value of tan 2pi/3 is given as -1.73205. . .. We can find the value of tan 2pi/3 by:

• Using Trigonometric Functions
• Using Unit Circle

Tan 2pi/3 in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 2pi/3 as:

• sin(2pi/3)/cos(2pi/3)
• ± sin(2pi/3)/√(1 - sin²(2pi/3))
• ± √(1 - cos²(2pi/3))/cos(2pi/3)
• ± 1/√(cosec²(2pi/3) - 1)
• ± √(sec²(2pi/3) - 1)
• 1/cot(2pi/3)

Note: Since 2pi/3 lies in the 2nd Quadrant, the final value of tan 2pi/3 will be negative.

We can use trigonometric identities to represent tan 2pi/3 as,

• cot(pi/2 - 2pi/3) = cot(-pi/6)
• -cot(pi/2 + 2pi/3) = -cot 7pi/6
• -tan (pi - 2pi/3) = -tan pi/3

Tan 2pi/3 Using Unit Circle

To find the value of tan 2π/3 using the unit circle:

• Rotate ‘r’ anticlockwise to form 2pi/3 angle with the positive x-axis.
• The tan of 2pi/3 equals the y-coordinate(0.866) divided by the x-coordinate(-0.5) of the point of intersection (-0.5, 0.866) of unit circle and r.

Hence the value of tan 2pi/3 = y/x = -1.7321 (approx)

☛ Also Check:

• sin 7pi/12
• cos pi/4
• cot 4pi/3
• cos 9pi/2
• cos pi/2
• cos 11pi/6

FAQs on Tan 2pi/3

What is Tan 2pi/3?

Tan 2pi/3 is the value of tangent trigonometric function for an angle equal to 2π/3 radians. The value of tan 2pi/3 is -√3 or -1.7321 (approx).

What is the Value of Tan 2pi/3 in Terms of Cos 2pi/3?

We know, using trig identities, we can write tan 2pi/3 as √(1 - cos²(2pi/3))/cos 2pi/3. Here, the value of cos 2pi/3 is equal to -0.5.

How to Find Tan 2pi/3 in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 2pi/3 can be given in terms of other trigonometric functions as:

• sin(2pi/3)/cos(2pi/3)
• ± sin(2pi/3)/√(1 - sin²(2pi/3))
• ± √(1 - cos²(2pi/3))/cos(2pi/3)
• ± 1/√(cosec²(2pi/3) - 1)
• ± √(sec²(2pi/3) - 1)
• 1/cot(2pi/3)

☛ Also check: trigonometry table

How to Find the Value of Tan 2pi/3?

The value of tan 2pi/3 can be calculated by constructing an angle of 2π/3 radians with the x-axis, and then finding the coordinates of the corresponding point (-0.5, 0.866) on the unit circle. The value of tan 2pi/3 is equal to the y-coordinate(0.866) divided by the x-coordinate (-0.5). ∴ tan 2pi/3 = -√3 or -1.7321

What is the Exact Value of tan 2pi/3?

The exact value of tan 2pi/3 can be given accurately up to 8 decimal places as -1.73205080 or as -√3.