from a handpicked tutor in LIVE 1to1 classes
Sin pi/4
The value of sin pi/4 is 0.7071067. . .. Sin pi/4 radians in degrees is written as sin ((π/4) × 180°/π), i.e., sin (45°). In this article, we will discuss the methods to find the value of sin pi/4 with examples.
 Sin pi/4: 1/√2
 Sin pi/4 in decimal: 0.7071067. . .
 Sin (pi/4): 0.7071067. . . or (1/√2)
 Sin pi/4 in degrees: sin (45°)
What is the Value of Sin pi/4?
The value of sin pi/4 in decimal is 0.707106781. . .. Sin pi/4 can also be expressed using the equivalent of the given angle (pi/4) in degrees (45°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ pi/4 radians = pi/4 × (180°/pi) = 45° or 45 degrees
∴ sin pi/4 = sin π/4 = sin(45°) = 1/√2 or 0.7071067. . .
Explanation:
For sin pi/4, the angle pi/4 lies between 0 and pi/2 (First Quadrant). Since sine function is positive in the first quadrant, thus sin pi/4 value = 1/√2 or 0.7071067. . .
Since the sine function is a periodic function, we can represent sin pi/4 as, sin pi/4 = sin(pi/4 + n × 2pi), n ∈ Z.
⇒ sin pi/4 = sin 9pi/4 = sin 17pi/4 , and so on.
Note: Since, sine is an odd function, the value of sin(pi/4) = sin(pi/4).
Methods to Find Value of Sin pi/4
The sine function is positive in the 1st quadrant. The value of sin pi/4 is given as 0.70710. . .. We can find the value of sin pi/4 by:
 Using Unit Circle
 Using Trigonometric Functions
Sin pi/4 Using Unit Circle
To find the value of sin π/4 using the unit circle:
 Rotate ‘r’ anticlockwise to form pi/4 angle with the positive xaxis.
 The sin of pi/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 pi/4 = y = 0.7071 (approx)
Sin pi/4 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin pi/4 as:
 ± √(1cos²(pi/4))
 ± tan(pi/4)/√(1 + tan²(pi/4))
 ± 1/√(1 + cot²(pi/4))
 ± √(sec²(pi/4)  1)/sec(pi/4)
 1/cosec(pi/4)
Note: Since pi/4 lies in the 1st Quadrant, the final value of sin pi/4 will be positive.
We can use trigonometric identities to represent sin pi/4 as,
 sin(pi  pi/4) = sin 3pi/4
 sin(pi + pi/4) = sin 5pi/4
 cos(pi/2  pi/4) = cos pi/4
 cos(pi/2 + pi/4) = cos 3pi/4
☛ Also Check:
Examples Using Sin pi/4

Example 1: Simplify: 6 (sin(pi/4)/sin(9pi/4))
Solution:
We know sin pi/4 = sin 9pi/4
⇒ 6 sin(pi/4)/sin(9pi/4) = 6(sin(pi/4)/sin(pi/4))
= 6(1) = 6 
Example 2: Using the value of sin pi/4, solve: (1cos²(pi/4)).
Solution:
We know, (1cos²(pi/4)) = (sin²(pi/4)) = 0.5
⇒ (1cos²(pi/4)) = 0.5 
Example 3: Find the value of 5 sin(pi/4)/7 cos(pi/4).
Solution:
Using trigonometric identities, we know, sin(pi/4) = cos(pi/2  pi/4) = cos(pi/4).
⇒ sin(pi/4) = cos(pi/4)
⇒ Value of 5 sin(pi/4)/7 cos(pi/4) = 5/7
FAQs on Sin pi/4
What is Sin pi/4?
Sin pi/4 is the value of sine trigonometric function for an angle equal to pi/4 radians. The value of sin pi/4 is 1/√2 or 0.7071 (approx).
How to Find Sin pi/4 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin π/4 can be given in terms of other trigonometric functions as:
 ± √(1cos²(pi/4))
 ± tan(pi/4)/√(1 + tan²(pi/4))
 ± 1/√(1 + cot²(pi/4))
 ± √(sec²(pi/4)  1)/sec(pi/4)
 1/cosec(pi/4)
☛ Also check: trigonometric table
What is the Value of Sin pi/4 in Terms of Cot pi/4?
We can represent the sine function in terms of the cotangent function using trig identities, sin pi/4 can be written as 1/√(1 + cot²(pi/4)). Here, the value of cot pi/4 is equal to 1.
What is the Value of Sin pi/4 in Terms of Cosec pi/4?
Since the cosecant function is the reciprocal of the sine function, we can write sin pi/4 as 1/cosec(pi/4). The value of cosec pi/4 is equal to 1.41421.
How to Find the Value of Sin pi/4?
The value of sin pi/4 can be calculated by constructing an angle of π/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 pi/4 is equal to the ycoordinate (0.7071). ∴ sin pi/4 = 0.7071.
visual curriculum