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 x-axis.
• The tan of pi/8 equals the y-coordinate(0.3827) divided by the x-coordinate(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:

• cos 11pi/6
• cos pi/6
• sin pi/12
• tan 7pi/4
• sin 2pi
• sin 11pi/6

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 x-axis, 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 y-coordinate(0.3827) divided by the x-coordinate (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