Tan 4pi/3

The value of tan 4pi/3 is 1.7320508. . .. Tan 4pi/3 radians in degrees is written as tan ((4π/3) × 180°/π), i.e., tan (240°). In this article, we will discuss the methods to find the value of tan 4pi/3 with examples.

• Tan 4pi/3: √3
• Tan 4pi/3 in decimal: 1.7320508. . .
• Tan (-4pi/3): -1.7320508. . . or -√3
• Tan 4pi/3 in degrees: tan (240°)

What is the Value of Tan 4pi/3?

The value of tan 4pi/3 in decimal is 1.732050807. . .. Tan 4pi/3 can also be expressed using the equivalent of the given angle (4pi/3) in degrees (240°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 4pi/3 radians = 4pi/3 × (180°/pi) = 240° or 240 degrees
∴ tan 4pi/3 = tan 4π/3 = tan(240°) = √3 or 1.7320508. . .

Explanation:

For tan 4pi/3, the angle 4pi/3 lies between pi and 3pi/2 (Third Quadrant). Since tangent function is positive in the third quadrant, thus tan 4pi/3 value = √3 or 1.7320508. . .
Since the tangent function is a periodic function, we can represent tan 4pi/3 as, tan 4pi/3 = tan(4pi/3 + n × pi), n ∈ Z.
⇒ tan 4pi/3 = tan 7pi/3 = tan 10pi/3 , and so on.
Note: Since, tangent is an odd function, the value of tan(-4pi/3) = -tan(4pi/3).

Methods to Find Value of Tan 4pi/3

The tangent function is positive in the 3rd quadrant. The value of tan 4pi/3 is given as 1.73205. . .. We can find the value of tan 4pi/3 by:

• Using Unit Circle
• Using Trigonometric Functions

Tan 4pi/3 Using Unit Circle

To find the value of tan 4π/3 using the unit circle:

• Rotate ‘r’ anticlockwise to form 4pi/3 angle with the positive x-axis.
• The tan of 4pi/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 4pi/3 = y/x = 1.7321 (approx)

Tan 4pi/3 in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 4pi/3 as:

• sin(4pi/3)/cos(4pi/3)
• ± sin(4pi/3)/√(1 - sin²(4pi/3))
• ± √(1 - cos²(4pi/3))/cos(4pi/3)
• ± 1/√(cosec²(4pi/3) - 1)
• ± √(sec²(4pi/3) - 1)
• 1/cot(4pi/3)

Note: Since 4pi/3 lies in the 3rd Quadrant, the final value of tan 4pi/3 will be positive.

We can use trigonometric identities to represent tan 4pi/3 as,

• cot(pi/2 - 4pi/3) = cot(-5pi/6)
• -cot(pi/2 + 4pi/3) = -cot 11pi/6
• -tan (pi - 4pi/3) = -tan(-pi/3)

☛ Also Check:

• cos 2pi/8
• csc pi/2
• sin 2pi/3
• sin pi/3
• sin 5pi/12
• cos pi/6

FAQs on Tan 4pi/3

What is Tan 4pi/3?

Tan 4pi/3 is the value of tangent trigonometric function for an angle equal to 4π/3 radians. The value of tan 4pi/3 is √3 or 1.7321 (approx).

How to Find Tan 4pi/3 in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 4pi/3 can be given in terms of other trigonometric functions as:

• sin(4pi/3)/cos(4pi/3)
• ± sin(4pi/3)/√(1 - sin²(4pi/3))
• ± √(1 - cos²(4pi/3))/cos(4pi/3)
• ± 1/√(cosec²(4pi/3) - 1)
• ± √(sec²(4pi/3) - 1)
• 1/cot(4pi/3)

☛ Also check: trigonometric table

What is the Value of Tan 4pi/3 in Terms of Cot 4pi/3?

Since the tangent function is the reciprocal of the cotangent function, we can write tan 4pi/3 as 1/cot(4pi/3). The value of cot 4pi/3 is equal to 1/√3.

What is the Value of Tan 4pi/3 in Terms of Cosec 4pi/3?

Since the tangent function can be represented using the cosecant function, we can write tan 4pi/3 as 1/√(cosec²(4pi/3) - 1). The value of cosec 4pi/3 is equal to -1.15470.

How to Find the Value of Tan 4pi/3?

The value of tan 4pi/3 can be calculated by constructing an angle of 4π/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 4pi/3 is equal to the y-coordinate(-0.866) divided by the x-coordinate (-0.5). ∴ tan 4pi/3 = √3 or 1.7321