Sin pi/8
The value of sin pi/8 is 0.3826834. . .. Sin pi/8 radians in degrees is written as sin ((π/8) × 180°/π), i.e., sin (22.5°). In this article, we will discuss the methods to find the value of sin pi/8 with examples.
 Sin pi/8 in decimal: 0.3826834. . .
 Sin (pi/8): 0.3826834. . .
 Sin pi/8 in degrees: sin (22.5°)
What is the Value of Sin pi/8?
The value of sin pi/8 in decimal is 0.382683432. . .. Sin 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
∴ sin pi/8 = sin π/8 = sin(22.5°) = 0.3826834. . .
Explanation:
For sin pi/8, the angle pi/8 lies between 0 and pi/2 (First Quadrant). Since sine function is positive in the first quadrant, thus sin pi/8 value = 0.3826834. . .
Since the sine function is a periodic function, we can represent sin pi/8 as, sin pi/8 = sin(pi/8 + n × 2pi), n ∈ Z.
⇒ sin pi/8 = sin 17pi/8 = sin 33pi/8 , and so on.
Note: Since, sine is an odd function, the value of sin(pi/8) = sin(pi/8).
Methods to Find Value of Sin pi/8
The sine function is positive in the 1st quadrant. The value of sin pi/8 is given as 0.38268. . .. We can find the value of sin pi/8 by:
 Using Trigonometric Functions
 Using Unit Circle
Sin pi/8 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin pi/8 as:
 ± √(1cos²(pi/8))
 ± tan(pi/8)/√(1 + tan²(pi/8))
 ± 1/√(1 + cot²(pi/8))
 ± √(sec²(pi/8)  1)/sec(pi/8)
 1/cosec(pi/8)
Note: Since pi/8 lies in the 1st Quadrant, the final value of sin pi/8 will be positive.
We can use trigonometric identities to represent sin pi/8 as,
 sin(pi  pi/8) = sin 7pi/8
 sin(pi + pi/8) = sin 9pi/8
 cos(pi/2  pi/8) = cos 3pi/8
 cos(pi/2 + pi/8) = cos 5pi/8
Sin pi/8 Using Unit Circle
To find the value of sin π/8 using the unit circle:
 Rotate ‘r’ anticlockwise to form pi/8 angle with the positive xaxis.
 The sin of pi/8 equals the ycoordinate(0.3827) of the point of intersection (0.9239, 0.3827) of unit circle and r.
Hence the value of sin pi/8 = y = 0.3827 (approx)
☛ Also Check:
Examples Using Sin pi/8

Example 1: Simplify: 6 (sin(pi/8)/sin(17pi/8))
Solution:
We know sin pi/8 = sin 17pi/8
⇒ 6 sin(pi/8)/sin(17pi/8) = 6(sin(pi/8)/sin(pi/8))
= 6(1) = 6 
Example 2: Find the value of sin(pi/8) if cosec(pi/8) is 2.6131.
Solution:
Since, sin pi/8 = 1/csc(pi/8)
⇒ sin pi/8 = 1/2.6131 = 0.3827 
Example 3: Find the value of 2 × (sin(pi/16) cos(pi/16)). [Hint: Use sin pi/8 = 0.3827]
Solution:
Using the sin 2a formula,
2 sin(pi/16) cos(pi/16) = sin(2 × pi/16) = sin pi/8
∵ sin pi/8 = 0.3827
⇒ 2 × (sin(pi/16) cos(pi/16)) = 0.3827
FAQs on Sin pi/8
What is Sin pi/8?
Sin pi/8 is the value of sine trigonometric function for an angle equal to pi/8 radians. The value of sin pi/8 is 0.3827 (approx).
What is the Exact Value of sin pi/8?
The exact value of sin pi/8 can be given accurately up to 8 decimal places as 0.38268343.
What is the Value of Sin pi/8 in Terms of Tan pi/8?
We know, using trig identities, we can write sin pi/8 as tan(pi/8)/√(1 + tan²(pi/8)). Here, the value of tan pi/8 is equal to 0.414213.
How to Find Sin pi/8 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin π/8 can be given in terms of other trigonometric functions as:
 ± √(1cos²(pi/8))
 ± tan(pi/8)/√(1 + tan²(pi/8))
 ± 1/√(1 + cot²(pi/8))
 ± √(sec²(pi/8)  1)/sec(pi/8)
 1/cosec(pi/8)
☛ Also check: trigonometry table
How to Find the Value of Sin pi/8?
The value of sin 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 sin pi/8 is equal to the ycoordinate (0.3827). ∴ sin pi/8 = 0.3827.
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