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 xaxis.
 The tan of 3pi/8 equals the ycoordinate(0.9239) divided by the xcoordinate(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:
Examples Using Tan 3pi/8

Example 1: Find the value of 2 tan(3pi/16)/(1  tan²(3pi/16)). [Hint: Use tan 3pi/8 = 2.4142]
Solution:
Using the tan 2a formula,
2 tan(3pi/16)/(1  tan²(3pi/16)) = tan(2 × 3pi/16) = tan 3pi/8
∵ tan 3pi/8 = 2.4142
⇒ 2 tan(3pi/16)/(1  tan²(3pi/16)) = 2.4142 
Example 2: Find the value of 4 tan(3pi/8)/10 tan(5pi/8).
Solution:
Using trigonometric identities, we know, tan(3pi/8) = tan(pi  3pi/8) = tan 5pi/8.
⇒ tan(3pi/8) = tan(5pi/8)
⇒ Value of 4 tan(3pi/8)/10 tan(5pi/8) = 4/10 = 2/5 
Example 3: Using the value of tan 3pi/8, solve: (sec²(3pi/8)  1).
Solution:
We know, (sec²(3pi/8)  1) = (tan²(3pi/8)) = 5.8284
⇒ (sec²(3pi/8)  1) = 5.8284
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 xaxis, 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 ycoordinate(0.9239) divided by the xcoordinate (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.
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