from a handpicked tutor in LIVE 1to1 classes
Tan pi/8
The value of tan pi/8 is 0.4142135. . .. Tan pi/8 radians in degrees is written as tan ((π/8) × 180°/π), i.e., tan (22.5°). In this article, we will discuss the methods to find the value of tan pi/8 with examples.
 Tan pi/8 in decimal: 0.4142135. . .
 Tan (pi/8): 0.4142135. . .
 Tan pi/8 in degrees: tan (22.5°)
What is the Value of Tan pi/8?
The value of tan pi/8 in decimal is 0.414213562. . .. Tan pi/8 can also be expressed using the equivalent of the given angle (pi/8) in degrees (22.5°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ pi/8 radians = pi/8 × (180°/pi) = 22.5° or 22.5 degrees
∴ tan pi/8 = tan π/8 = tan(22.5°) = 0.4142135. . .
Explanation:
For tan pi/8, the angle pi/8 lies between 0 and pi/2 (First Quadrant). Since tangent function is positive in the first quadrant, thus tan pi/8 value = 0.4142135. . .
Since the tangent function is a periodic function, we can represent tan pi/8 as, tan pi/8 = tan(pi/8 + n × pi), n ∈ Z.
⇒ tan pi/8 = tan 9pi/8 = tan 17pi/8 , and so on.
Note: Since, tangent is an odd function, the value of tan(pi/8) = tan(pi/8).
Methods to Find Value of Tan pi/8
The tangent function is positive in the 1st quadrant. The value of tan pi/8 is given as 0.41421. . .. We can find the value of tan pi/8 by:
 Using Trigonometric Functions
 Using Unit Circle
Tan pi/8 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the tan pi/8 as:
 sin(pi/8)/cos(pi/8)
 ± sin(pi/8)/√(1  sin²(pi/8))
 ± √(1  cos²(pi/8))/cos(pi/8)
 ± 1/√(cosec²(pi/8)  1)
 ± √(sec²(pi/8)  1)
 1/cot(pi/8)
Note: Since pi/8 lies in the 1st Quadrant, the final value of tan pi/8 will be positive.
We can use trigonometric identities to represent tan pi/8 as,
 cot(pi/2  pi/8) = cot 3pi/8
 cot(pi/2 + pi/8) = cot 5pi/8
 tan (pi  pi/8) = tan 7pi/8
Tan pi/8 Using Unit Circle
To find the value of tan π/8 using the unit circle:
 Rotate ‘r’ anticlockwise to form pi/8 angle with the positive xaxis.
 The tan of pi/8 equals the ycoordinate(0.3827) divided by the xcoordinate(0.9239) of the point of intersection (0.9239, 0.3827) of unit circle and r.
Hence the value of tan pi/8 = y/x = 0.4142 (approx)
☛ Also Check:
Examples Using Tan pi/8

Example 1: Simplify: 3 (tan(pi/8)/cot(3pi/8))
Solution:
We know tan pi/8 = cot 3pi/8
⇒ 3 tan(pi/8)/cot(3pi/8) = 3 tan(pi/8)/tan(pi/8)
= 3(1) = 3 
Example 2: Find the value of 2 tan(pi/8)/8 tan(7pi/8).
Solution:
Using trigonometric identities, we know, tan(pi/8) = tan(pi  pi/8) = tan 7pi/8.
⇒ tan(pi/8) = tan(7pi/8)
⇒ Value of 2 tan(pi/8)/8 tan(7pi/8) = 2/8 = 1/4 
Example 3: Find the value of tan pi/8 if cot pi/8 is 2.4142.
Solution:
Since, tan pi/8 = 1/cot(pi/8)
⇒ tan pi/8 = 1/2.4142 = 0.4142
FAQs on Tan pi/8
What is Tan pi/8?
Tan pi/8 is the value of tangent trigonometric function for an angle equal to π/8 radians. The value of tan pi/8 is 0.4142 (approx).
How to Find the Value of Tan pi/8?
The value of tan pi/8 can be calculated by constructing an angle of π/8 radians with the xaxis, and then finding the coordinates of the corresponding point (0.9239, 0.3827) on the unit circle. The value of tan pi/8 is equal to the ycoordinate(0.3827) divided by the xcoordinate (0.9239). ∴ tan pi/8 = 0.4142
What is the Exact Value of tan pi/8?
The exact value of tan pi/8 can be given accurately up to 8 decimal places as 0.41421356.
What is the Value of Tan pi/8 in Terms of Cos pi/8?
We know, using trig identities, we can write tan pi/8 as √(1  cos²(pi/8))/cos pi/8. Here, the value of cos pi/8 is equal to 0.923879.
How to Find Tan pi/8 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of tan pi/8 can be given in terms of other trigonometric functions as:
 sin(pi/8)/cos(pi/8)
 ± sin(pi/8)/√(1  sin²(pi/8))
 ± √(1  cos²(pi/8))/cos(pi/8)
 ± 1/√(cosec²(pi/8)  1)
 ± √(sec²(pi/8)  1)
 1/cot(pi/8)
☛ Also check: trigonometric table
visual curriculum