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Sin 5pi/4
The value of sin 5pi/4 is 0.7071067. . .. Sin 5pi/4 radians in degrees is written as sin ((5π/4) × 180°/π), i.e., sin (225°). In this article, we will discuss the methods to find the value of sin 5pi/4 with examples.
 Sin 5pi/4: (1/√2)
 Sin 5pi/4 in decimal: 0.7071067. . .
 Sin (5pi/4): 0.7071067. . . or 1/√2
 Sin 5pi/4 in degrees: sin (225°)
What is the Value of Sin 5pi/4?
The value of sin 5pi/4 in decimal is 0.707106781. . .. Sin 5pi/4 can also be expressed using the equivalent of the given angle (5pi/4) in degrees (225°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 5pi/4 radians = 5pi/4 × (180°/pi) = 225° or 225 degrees
∴ sin 5pi/4 = sin 5π/4 = sin(225°) = (1/√2) or 0.7071067. . .
Explanation:
For sin 5pi/4, the angle 5pi/4 lies between pi and 3pi/2 (Third Quadrant). Since sine function is negative in the third quadrant, thus sin 5pi/4 value = (1/√2) or 0.7071067. . .
Since the sine function is a periodic function, we can represent sin 5pi/4 as, sin 5pi/4 = sin(5pi/4 + n × 2pi), n ∈ Z.
⇒ sin 5pi/4 = sin 13pi/4 = sin 21pi/4 , and so on.
Note: Since, sine is an odd function, the value of sin(5pi/4) = sin(5pi/4).
Methods to Find Value of Sin 5pi/4
The sine function is negative in the 3rd quadrant. The value of sin 5pi/4 is given as 0.70710. . .. We can find the value of sin 5pi/4 by:
 Using Trigonometric Functions
 Using Unit Circle
Sin 5pi/4 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 5pi/4 as:
 ± √(1cos²(5pi/4))
 ± tan(5pi/4)/√(1 + tan²(5pi/4))
 ± 1/√(1 + cot²(5pi/4))
 ± √(sec²(5pi/4)  1)/sec(5pi/4)
 1/cosec(5pi/4)
Note: Since 5pi/4 lies in the 3rd Quadrant, the final value of sin 5pi/4 will be negative.
We can use trigonometric identities to represent sin 5pi/4 as,
 sin(pi  5pi/4) = sin(pi/4)
 sin(pi + 5pi/4) = sin 9pi/4
 cos(pi/2  5pi/4) = cos(3pi/4)
 cos(pi/2 + 5pi/4) = cos 7pi/4
Sin 5pi/4 Using Unit Circle
To find the value of sin 5π/4 using the unit circle:
 Rotate ‘r’ anticlockwise to form 5pi/4 angle with the positive xaxis.
 The sin of 5pi/4 equals the ycoordinate(0.7071) of the point of intersection (0.7071, 0.7071) of unit circle and r.
Hence the value of sin 5pi/4 = y = 0.7071 (approx)
☛ Also Check:
Examples Using Sin 5pi/4

Example 1: Find the value of 2 × (sin(5pi/8) cos(5pi/8)). [Hint: Use sin 5pi/4 = 0.7071]
Solution:
Using the sin 2a formula,
2 sin(5pi/8) cos(5pi/8) = sin(2 × 5pi/8) = sin 5pi/4
∵ sin 5pi/4 = 0.7071
⇒ 2 × (sin(5pi/8) cos(5pi/8)) = 0.7071 
Example 2: Simplify: 7 (sin(5pi/4)/sin(13pi/4))
Solution:
We know sin 5pi/4 = sin 13pi/4
⇒ 7 sin(5pi/4)/sin(13pi/4) = 7(sin(5pi/4)/sin(5pi/4))
= 7(1) = 7 
Example 3: Find the value of sin(5pi/4) if cosec(5pi/4) is 1.4142.
Solution:
Since, sin 5pi/4 = 1/csc(5pi/4)
⇒ sin 5pi/4 = 1/(1.4142) = 0.7071
FAQs on Sin 5pi/4
What is Sin 5pi/4?
Sin 5pi/4 is the value of sine trigonometric function for an angle equal to 5pi/4 radians. The value of sin 5pi/4 is (1/√2) or 0.7071 (approx).
What is the Value of Sin 5pi/4 in Terms of Sec 5pi/4?
Since the sine function can be represented using the secant function, we can write sin 5pi/4 as √(sec²(5pi/4)  1)/sec 5pi/4. The value of sec 5pi/4 is equal to 1.414213.
How to Find Sin 5pi/4 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 5π/4 can be given in terms of other trigonometric functions as:
 ± √(1cos²(5pi/4))
 ± tan(5pi/4)/√(1 + tan²(5pi/4))
 ± 1/√(1 + cot²(5pi/4))
 ± √(sec²(5pi/4)  1)/sec(5pi/4)
 1/cosec(5pi/4)
☛ Also check: trigonometric table
What is the Value of Sin 5pi/4 in Terms of Tan 5pi/4?
We know, using trig identities, we can write sin 5pi/4 as tan(5pi/4)/√(1 + tan²(5pi/4)). Here, the value of tan 5pi/4 is equal to 1.
How to Find the Value of Sin 5pi/4?
The value of sin 5pi/4 can be calculated by constructing an angle of 5π/4 radians with the xaxis, and then finding the coordinates of the corresponding point (0.7071, 0.7071) on the unit circle. The value of sin 5pi/4 is equal to the ycoordinate (0.7071). ∴ sin 5pi/4 = 0.7071.
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