Sin 3pi/4
The value of sin 3pi/4 is 0.7071067. . .. Sin 3pi/4 radians in degrees is written as sin ((3π/4) × 180°/π), i.e., sin (135°). In this article, we will discuss the methods to find the value of sin 3pi/4 with examples.
 Sin 3pi/4: 1/√2
 Sin 3pi/4 in decimal: 0.7071067. . .
 Sin (3pi/4): 0.7071067. . . or (1/√2)
 Sin 3pi/4 in degrees: sin (135°)
What is the Value of Sin 3pi/4?
The value of sin 3pi/4 in decimal is 0.707106781. . .. Sin 3pi/4 can also be expressed using the equivalent of the given angle (3pi/4) in degrees (135°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 3pi/4 radians = 3pi/4 × (180°/pi) = 135° or 135 degrees
∴ sin 3pi/4 = sin 3π/4 = sin(135°) = 1/√2 or 0.7071067. . .
Explanation:
For sin 3pi/4, the angle 3pi/4 lies between pi/2 and pi (Second Quadrant). Since sine function is positive in the second quadrant, thus sin 3pi/4 value = 1/√2 or 0.7071067. . .
Since the sine function is a periodic function, we can represent sin 3pi/4 as, sin 3pi/4 = sin(3pi/4 + n × 2pi), n ∈ Z.
⇒ sin 3pi/4 = sin 11pi/4 = sin 19pi/4 , and so on.
Note: Since, sine is an odd function, the value of sin(3pi/4) = sin(3pi/4).
Methods to Find Value of Sin 3pi/4
The sine function is positive in the 2nd quadrant. The value of sin 3pi/4 is given as 0.70710. . .. We can find the value of sin 3pi/4 by:
 Using Trigonometric Functions
 Using Unit Circle
Sin 3pi/4 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 3pi/4 as:
 ± √(1cos²(3pi/4))
 ± tan(3pi/4)/√(1 + tan²(3pi/4))
 ± 1/√(1 + cot²(3pi/4))
 ± √(sec²(3pi/4)  1)/sec(3pi/4)
 1/cosec(3pi/4)
Note: Since 3pi/4 lies in the 2nd Quadrant, the final value of sin 3pi/4 will be positive.
We can use trigonometric identities to represent sin 3pi/4 as,
 sin(pi  3pi/4) = sin pi/4
 sin(pi + 3pi/4) = sin 7pi/4
 cos(pi/2  3pi/4) = cos(pi/4)
 cos(pi/2 + 3pi/4) = cos 5pi/4
Sin 3pi/4 Using Unit Circle
To find the value of sin 3π/4 using the unit circle:
 Rotate ‘r’ anticlockwise to form 3pi/4 angle with the positive xaxis.
 The sin of 3pi/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 3pi/4 = y = 0.7071 (approx)
☛ Also Check:
Examples Using Sin 3pi/4

Example 1: Find the value of 2 × (sin(3pi/8) cos(3pi/8)). [Hint: Use sin 3pi/4 = 0.7071]
Solution:
Using the sin 2a formula,
2 sin(3pi/8) cos(3pi/8) = sin(2 × 3pi/8) = sin 3pi/4
∵ sin 3pi/4 = 0.7071
⇒ 2 × (sin(3pi/8) cos(3pi/8)) = 0.7071 
Example 2: Find the value of 5 sin(3pi/4)/7 cos(pi/4).
Solution:
Using trigonometric identities, we know, sin(3pi/4) = cos(pi/2  3pi/4) = cos(pi/4).
⇒ sin(3pi/4) = cos(pi/4)
⇒ Value of 5 sin(3pi/4)/7 cos(pi/4) = 5/7 
Example 3: Using the value of sin 3pi/4, solve: (1cos²(3pi/4)).
Solution:
We know, (1cos²(3pi/4)) = (sin²(3pi/4)) = 0.5
⇒ (1cos²(3pi/4)) = 0.5
FAQs on Sin 3pi/4
What is Sin 3pi/4?
Sin 3pi/4 is the value of sine trigonometric function for an angle equal to 3pi/4 radians. The value of sin 3pi/4 is 1/√2 or 0.7071 (approx).
How to Find the Value of Sin 3pi/4?
The value of sin 3pi/4 can be calculated by constructing an angle of 3π/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 3pi/4 is equal to the ycoordinate (0.7071). ∴ sin 3pi/4 = 0.7071.
What is the Value of Sin 3pi/4 in Terms of Cosec 3pi/4?
Since the cosecant function is the reciprocal of the sine function, we can write sin 3pi/4 as 1/cosec(3pi/4). The value of cosec 3pi/4 is equal to 1.41421.
How to Find Sin 3pi/4 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 3π/4 can be given in terms of other trigonometric functions as:
 ± √(1cos²(3pi/4))
 ± tan(3pi/4)/√(1 + tan²(3pi/4))
 ± 1/√(1 + cot²(3pi/4))
 ± √(sec²(3pi/4)  1)/sec(3pi/4)
 1/cosec(3pi/4)
☛ Also check: trigonometric table
What is the Value of Sin 3pi/4 in Terms of Cos 3pi/4?
Using trigonometric identities, we can write sin 3pi/4 in terms of cos 3pi/4 as, sin(3pi/4) = √(1cos²(3pi/4)). Here, the value of cos 3pi/4 is equal to −(1/√2) .
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