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Sin 5pi/8
The value of sin 5pi/8 is 0.9238795. . .. Sin 5pi/8 radians in degrees is written as sin ((5π/8) × 180°/π), i.e., sin (112.5°). In this article, we will discuss the methods to find the value of sin 5pi/8 with examples.
 Sin 5pi/8 in decimal: 0.9238795. . .
 Sin (5pi/8): 0.9238795. . .
 Sin 5pi/8 in degrees: sin (112.5°)
What is the Value of Sin 5pi/8?
The value of sin 5pi/8 in decimal is 0.923879532. . .. Sin 5pi/8 can also be expressed using the equivalent of the given angle (5pi/8) in degrees (112.5°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 5pi/8 radians = 5pi/8 × (180°/pi) = 112.5° or 112.5 degrees
∴ sin 5pi/8 = sin 5π/8 = sin(112.5°) = 0.9238795. . .
Explanation:
For sin 5pi/8, the angle 5pi/8 lies between pi/2 and pi (Second Quadrant). Since sine function is positive in the second quadrant, thus sin 5pi/8 value = 0.9238795. . .
Since the sine function is a periodic function, we can represent sin 5pi/8 as, sin 5pi/8 = sin(5pi/8 + n × 2pi), n ∈ Z.
⇒ sin 5pi/8 = sin 21pi/8 = sin 37pi/8 , and so on.
Note: Since, sine is an odd function, the value of sin(5pi/8) = sin(5pi/8).
Methods to Find Value of Sin 5pi/8
The sine function is positive in the 2nd quadrant. The value of sin 5pi/8 is given as 0.92387. . .. We can find the value of sin 5pi/8 by:
 Using Trigonometric Functions
 Using Unit Circle
Sin 5pi/8 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 5pi/8 as:
 ± √(1cos²(5pi/8))
 ± tan(5pi/8)/√(1 + tan²(5pi/8))
 ± 1/√(1 + cot²(5pi/8))
 ± √(sec²(5pi/8)  1)/sec(5pi/8)
 1/cosec(5pi/8)
Note: Since 5pi/8 lies in the 2nd Quadrant, the final value of sin 5pi/8 will be positive.
We can use trigonometric identities to represent sin 5pi/8 as,
 sin(pi  5pi/8) = sin 3pi/8
 sin(pi + 5pi/8) = sin 13pi/8
 cos(pi/2  5pi/8) = cos(pi/8)
 cos(pi/2 + 5pi/8) = cos 9pi/8
Sin 5pi/8 Using Unit Circle
To find the value of sin 5π/8 using the unit circle:
 Rotate ‘r’ anticlockwise to form 5pi/8 angle with the positive xaxis.
 The sin of 5pi/8 equals the ycoordinate(0.9239) of the point of intersection (0.3827, 0.9239) of unit circle and r.
Hence the value of sin 5pi/8 = y = 0.9239 (approx)
☛ Also Check:
Examples Using Sin 5pi/8

Example 1: Find the value of sin(5pi/8) if cosec(5pi/8) is 1.0823.
Solution:
Since, sin 5pi/8 = 1/csc(5pi/8)
⇒ sin 5pi/8 = 1/1.0823 = 0.9239 
Example 2: Find the value of 2 × (sin(5pi/16) cos(5pi/16)). [Hint: Use sin 5pi/8 = 0.9239]
Solution:
Using the sin 2a formula,
2 sin(5pi/16) cos(5pi/16) = sin(2 × 5pi/16) = sin 5pi/8
∵ sin 5pi/8 = 0.9239
⇒ 2 × (sin(5pi/16) cos(5pi/16)) = 0.9239 
Example 3: Simplify: 3 (sin(5pi/8)/sin(21pi/8))
Solution:
We know sin 5pi/8 = sin 21pi/8
⇒ 3 sin(5pi/8)/sin(21pi/8) = 3(sin(5pi/8)/sin(5pi/8))
= 3(1) = 3
FAQs on Sin 5pi/8
What is Sin 5pi/8?
Sin 5pi/8 is the value of sine trigonometric function for an angle equal to 5pi/8 radians. The value of sin 5pi/8 is 0.9239 (approx).
How to Find the Value of Sin 5pi/8?
The value of sin 5pi/8 can be calculated by constructing an angle of 5π/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 sin 5pi/8 is equal to the ycoordinate (0.9239). ∴ sin 5pi/8 = 0.9239.
What is the Value of Sin 5pi/8 in Terms of Cot 5pi/8?
We can represent the sine function in terms of the cotangent function using trig identities, sin 5pi/8 can be written as 1/√(1 + cot²(5pi/8)). Here, the value of cot 5pi/8 is equal to 0.4142.
What is the Value of Sin 5pi/8 in Terms of Cosec 5pi/8?
Since the cosecant function is the reciprocal of the sine function, we can write sin 5pi/8 as 1/cosec(5pi/8). The value of cosec 5pi/8 is equal to 1.08239.
How to Find Sin 5pi/8 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 5π/8 can be given in terms of other trigonometric functions as:
 ± √(1cos²(5pi/8))
 ± tan(5pi/8)/√(1 + tan²(5pi/8))
 ± 1/√(1 + cot²(5pi/8))
 ± √(sec²(5pi/8)  1)/sec(5pi/8)
 1/cosec(5pi/8)
☛ Also check: trigonometry table
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