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Tan 2pi

The value of tan 2pi is 0. Tan 2pi radians in degrees is written as tan ((2π) × 180°/π), i.e., tan (360°). In this article, we will discuss the methods to find the value of tan 2pi with examples.

Tan 2pi: 0
Tan (-2pi): 0
Tan 2pi in degrees: tan (360°)

What is the Value of Tan 2pi?

The value of tan 2pi is 0. Tan 2pi can also be expressed using the equivalent of the given angle (2pi) in degrees (360°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 2pi radians = 2pi × (180°/pi) = 360° or 360 degrees
∴ tan 2pi = tan 2π = tan(360°) = 0

Explanation:

We can represent tan 2pi as, tan(2pi mod 2pi) = tan(0). For tan 2pi, the angle 2pi lies on the positive x-axis. Thus, tan 2pi value = 0
Since the tangent function is a periodic function, we can represent tan 2pi as, tan 2pi = tan(2pi + n × pi), n ∈ Z.
⇒ tan 2pi = tan 3pi = tan 4pi , and so on.
Note: Since, tangent is an odd function, the value of tan(-2pi) = -tan(2pi) = 0.

Methods to Find Value of Tan 2pi

The value of tan 2pi is given as 0. We can find the value of tan 2pi by:

Using Trigonometric Functions
Using Unit Circle

Tan 2pi in Terms of Trigonometric Functions

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

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

Note: Since 2pi lies on the positive x-axis, the final value of tan 2pi is 0.

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

cot(pi/2 - 2pi) = cot(-3pi/2)
-cot(pi/2 + 2pi) = -cot 5pi/2
-tan (pi - 2pi) = -tan(-pi)

Tan 2pi Using Unit Circle

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

Rotate ‘r’ anticlockwise to form 0 or 2pi angle with the positive x-axis.
The tan of 2pi equals the y-coordinate(0) divided by the x-coordinate(1) of the point of intersection (1, 0) of unit circle and r.

Hence the value of tan 2pi = y/x = 0

☛ Also Check:

Examples Using Tan 2pi

Example 1: Find the value of tan 2pi if sin 2pi is 0 and cos 2pi is 1.

Solution:
Since, tan 2pi = sin 2pi/cos(2pi)
⇒ tan 2pi = 0
Example 2: Using the value of tan 2pi, solve: (sec²(2pi) - 1).

Solution:
We know, (sec²(2pi) - 1) = (tan²(2pi)) = 0
⇒ (sec²(2pi) - 1) = 0
Example 3: Simplify: 7 (tan(2pi)/cot(pi/4))

Solution:

We know tan 2pi = 0 and cot(pi/4) = 1
⇒ 7 tan(2pi)/cot(pi/4) = 0

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FAQs on Tan 2pi

What is Tan 2pi?

Tan 2pi is the value of tangent trigonometric function for an angle equal to 2π radians. The value of tan 2pi is 0.

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

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

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

☛ Also check: trigonometric table

What is the Exact Value of tan 2pi?

The exact value of tan 2pi is 0.

What is the Value of Tan 2pi in Terms of Sin 2pi?

Using trigonometric identities, we can write tan 2pi in terms of sin 2pi as, tan(2pi) = sin(2pi)/√(1 - sin²(2pi)) . Here, the value of sin 2pi is equal to 0.

How to Find the Value of Tan 2pi?

The value of tan 2pi can be calculated by constructing an angle of 2π radians with the x-axis, and then finding the coordinates of the corresponding point (1, 0) on the unit circle. The value of tan 2pi is equal to the y-coordinate(0) divided by the x-coordinate (1). ∴ tan 2pi = 0

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