Tan 3pi/8

The value of tan 3pi/8 is 2.4142135. . .. Tan 3pi/8 radians in degrees is written as tan ((3π/8) × 180°/π), i.e., tan (67.5°). In this article, we will discuss the methods to find the value of tan 3pi/8 with examples.

• Tan 3pi/8 in decimal: 2.4142135. . .
• Tan (-3pi/8): -2.4142135. . .
• Tan 3pi/8 in degrees: tan (67.5°)

What is the Value of Tan 3pi/8?

The value of tan 3pi/8 in decimal is 2.414213562. . .. Tan 3pi/8 can also be expressed using the equivalent of the given angle (3pi/8) in degrees (67.5°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 3pi/8 radians = 3pi/8 × (180°/pi) = 67.5° or 67.5 degrees
∴ tan 3pi/8 = tan 3π/8 = tan(67.5°) = 2.4142135. . .

Explanation:

For tan 3pi/8, the angle 3pi/8 lies between 0 and pi/2 (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 3pi/8 value = 2.4142135. . .
Since the tangent function is a periodic function, we can represent tan 3pi/8 as, tan 3pi/8 = tan(3pi/8 + n × pi), n ∈ Z.
⇒ tan 3pi/8 = tan 11pi/8 = tan 19pi/8 , and so on.
Note: Since, tangent is an odd function, the value of tan(-3pi/8) = -tan(3pi/8).

Methods to Find Value of Tan 3pi/8

The tangent function is positive in the 1st quadrant. The value of tan 3pi/8 is given as 2.41421. . .. We can find the value of tan 3pi/8 by:

• Using Unit Circle
• Using Trigonometric Functions

Tan 3pi/8 Using Unit Circle

To find the value of tan 3π/8 using the unit circle:

• Rotate ‘r’ anticlockwise to form 3pi/8 angle with the positive x-axis.
• The tan of 3pi/8 equals the y-coordinate(0.9239) divided by the x-coordinate(0.3827) of the point of intersection (0.3827, 0.9239) of unit circle and r.

Hence the value of tan 3pi/8 = y/x = 2.4142 (approx)

Tan 3pi/8 in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 3pi/8 as:

• sin(3pi/8)/cos(3pi/8)
• ± sin(3pi/8)/√(1 - sin²(3pi/8))
• ± √(1 - cos²(3pi/8))/cos(3pi/8)
• ± 1/√(cosec²(3pi/8) - 1)
• ± √(sec²(3pi/8) - 1)
• 1/cot(3pi/8)

Note: Since 3pi/8 lies in the 1st Quadrant, the final value of tan 3pi/8 will be positive.

We can use trigonometric identities to represent tan 3pi/8 as,

• cot(pi/2 - 3pi/8) = cot pi/8
• -cot(pi/2 + 3pi/8) = -cot 7pi/8
• -tan (pi - 3pi/8) = -tan 5pi/8

☛ Also Check:

• cot pi/6
• cot pi/4
• csc pi/3
• cos 7pi/12
• sin 2pi
• tan 5pi/2

FAQs on Tan 3pi/8

What is Tan 3pi/8?

Tan 3pi/8 is the value of tangent trigonometric function for an angle equal to 3π/8 radians. The value of tan 3pi/8 is 2.4142 (approx).

How to Find Tan 3pi/8 in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 3pi/8 can be given in terms of other trigonometric functions as:

• sin(3pi/8)/cos(3pi/8)
• ± sin(3pi/8)/√(1 - sin²(3pi/8))
• ± √(1 - cos²(3pi/8))/cos(3pi/8)
• ± 1/√(cosec²(3pi/8) - 1)
• ± √(sec²(3pi/8) - 1)
• 1/cot(3pi/8)

☛ Also check: trigonometry table

What is the Value of Tan 3pi/8 in Terms of Cot 3pi/8?

Since the tangent function is the reciprocal of the cotangent function, we can write tan 3pi/8 as 1/cot(3pi/8). The value of cot 3pi/8 is equal to 0.41421.

How to Find the Value of Tan 3pi/8?

The value of tan 3pi/8 can be calculated by constructing an angle of 3π/8 radians with the x-axis, and then finding the coordinates of the corresponding point (0.3827, 0.9239) on the unit circle. The value of tan 3pi/8 is equal to the y-coordinate(0.9239) divided by the x-coordinate (0.3827). ∴ tan 3pi/8 = 2.4142

What is the Exact Value of tan 3pi/8?

The exact value of tan 3pi/8 can be given accurately up to 8 decimal places as 2.41421356.