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Tan pi

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

Tan pi: 0
Tan (-pi): 0
Tan pi in degrees: tan (180°)

What is the Value of Tan pi?

The value of tan pi is 0. Tan pi can also be expressed using the equivalent of the given angle (pi) in degrees (180°).

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

Explanation:

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

Methods to Find Value of Tan pi

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

Using Unit Circle
Using Trigonometric Functions

Tan pi Using Unit Circle

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

Rotate ‘r’ anticlockwise to form pi angle with the positive x-axis.
The tan of pi 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 pi = y/x = 0

Tan pi in Terms of Trigonometric Functions

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

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

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

We can use trigonometric identities to represent tan pi as,

cot(pi/2 - pi) = cot(-pi/2)
-cot(pi/2 + pi) = -cot 3pi/2
-tan (pi - pi) = -tan 0

☛ Also Check:

Examples Using Tan pi

Example 1: Using the value of tan pi, solve: (sec²(pi) - 1).

Solution:
We know, (sec²(pi) - 1) = (tan²(pi)) = 0
⇒ (sec²(pi) - 1) = 0
Example 2: Find the value of 5 tan(pi)/7 tan(pi/4).

Solution:

Using trigonometric values, we know, tan(pi) = 0 and tan pi/4 = 1
⇒ Value of 5 tan(pi)/7 tan(pi/4) = 0
Example 3: Simplify: 7 (tan(pi)/cot(pi/4))

Solution:

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

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FAQs on Tan pi

What is Tan pi?

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

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

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

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

☛ Also check: trigonometric table

What is the Value of Tan pi in Terms of Cosec pi?

Since the tangent function can be represented using the cosecant function, we can write tan pi as 1/√(cosec²(pi) - 1).

What is the Value of Tan pi in Terms of Cot pi?

Since the tangent function is the reciprocal of the cotangent function, we can write tan pi as 1/cot(pi).

How to Find the Value of Tan pi?

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

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