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Sin 9pi/4
The value of sin 9pi/4 is 0.7071067. . .. Sin 9pi/4 radians in degrees is written as sin ((9π/4) × 180°/π), i.e., sin (405°). In this article, we will discuss the methods to find the value of sin 9pi/4 with examples.
 Sin 9pi/4: 1/√2
 Sin 9pi/4 in decimal: 0.7071067. . .
 Sin (9pi/4): 0.7071067. . . or (1/√2)
 Sin 9pi/4 in degrees: sin (405°)
What is the Value of Sin 9pi/4?
The value of sin 9pi/4 in decimal is 0.707106781. . .. Sin 9pi/4 can also be expressed using the equivalent of the given angle (9pi/4) in degrees (405°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 9pi/4 radians = 9pi/4 × (180°/pi) = 405° or 405 degrees
∴ sin 9pi/4 = sin 9π/4 = sin(405°) = 1/√2 or 0.7071067. . .
Explanation:
For sin 9pi/4, the angle 9pi/4 > 2pi. Given the periodic property of the sine function, we can represent it as sin(9pi/4 mod 2pi) = sin(pi/4). The angle 9pi/4, coterminal to angle pi/4, is located in the First Quadrant(Quadrant I).
Since sin function is positive in the 1st quadrant, thus sin 9pi/4 value = 1/√2 or 0.7071067. . .
Similarly, sin 9pi/4 can also be written as, sin 9pi/4 = (9pi/4 + n × 2pi), n ∈ Z.
⇒ sin 9pi/4 = sin 17pi/4 = sin 25pi/4, and so on.
Note: Since, sine is an odd function, the value of sin(9pi/4) = sin(9pi/4).
Methods to Find Value of Sin 9pi/4
The sine function is positive in the 1st quadrant. The value of sin 9pi/4 is given as 0.70710. . .. We can find the value of sin 9pi/4 by:
 Using Unit Circle
 Using Trigonometric Functions
Sin 9pi/4 Using Unit Circle
To find the value of sin 9π/4 using the unit circle, represent 9pi/4 in the form (1 × 2pi) + pi/4 [∵ 9pi/4>2pi] ∵ sine is a periodic function, sin 9pi/4 = sin pi/4.
 Rotate ‘r’ anticlockwise to form pi/4 or 9pi/4 angle with the positive xaxis.
 The sin of 9pi/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 9pi/4 = y = 0.7071 (approx)
Sin 9pi/4 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 9pi/4 as:
 ± √(1cos²(9pi/4))
 ± tan(9pi/4)/√(1 + tan²(9pi/4))
 ± 1/√(1 + cot²(9pi/4))
 ± √(sec²(9pi/4)  1)/sec(9pi/4)
 1/cosec(9pi/4)
Note: Since 9pi/4 lies in the 1st Quadrant, the final value of sin 9pi/4 will be positive.
We can use trigonometric identities to represent sin 9pi/4 as,
 sin(pi  9pi/4) = sin(5pi/4)
 sin(pi + 9pi/4) = sin 13pi/4
 cos(pi/2  9pi/4) = cos(7pi/4)
 cos(pi/2 + 9pi/4) = cos 11pi/4
☛ Also Check:
Examples Using Sin 9pi/4

Example 1: Find the value of 2 × (sin(9pi/8) cos(9pi/8)). [Hint: Use sin 9pi/4 = 0.7071]
Solution:
Using the sin 2a formula,
2 sin(9pi/8) cos(9pi/8) = sin(2 × 9pi/8) = sin 9pi/4
∵ sin 9pi/4 = 0.7071
⇒ 2 × (sin(9pi/8) cos(9pi/8)) = 0.7071 
Example 2: Simplify: 6 (sin(9pi/4)/sin(25pi/4))
Solution:
We know sin 9pi/4 = sin 25pi/4
⇒ 6 sin(9pi/4)/sin(25pi/4) = 6(sin(9pi/4)/sin(9pi/4))
= 6(1) = 6 
Example 3: Find the value of sin(9pi/4) if cosec(9pi/4) is 1.4142.
Solution:
Since, sin 9pi/4 = 1/csc(9pi/4)
⇒ sin 9pi/4 = 1/1.4142 = 0.7071
FAQs on Sin 9pi/4
What is Sin 9pi/4?
Sin 9pi/4 is the value of sine trigonometric function for an angle equal to 9pi/4 radians. The value of sin 9pi/4 is 1/√2 or 0.7071 (approx).
How to Find Sin 9pi/4 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 9π/4 can be given in terms of other trigonometric functions as:
 ± √(1cos²(9pi/4))
 ± tan(9pi/4)/√(1 + tan²(9pi/4))
 ± 1/√(1 + cot²(9pi/4))
 ± √(sec²(9pi/4)  1)/sec(9pi/4)
 1/cosec(9pi/4)
☛ Also check: trigonometry table
What is the Value of Sin 9pi/4 in Terms of Sec 9pi/4?
Since the sine function can be represented using the secant function, we can write sin 9pi/4 as √(sec²(9pi/4)  1)/sec 9pi/4. The value of sec 9pi/4 is equal to 1.414213.
What is the Value of Sin 9pi/4 in Terms of Tan 9pi/4?
We know, using trig identities, we can write sin 9pi/4 as tan(9pi/4)/√(1 + tan²(9pi/4)). Here, the value of tan 9pi/4 is equal to 1.
How to Find the Value of Sin 9pi/4?
The value of sin 9pi/4 can be calculated by constructing an angle of 9π/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 9pi/4 is equal to the ycoordinate (0.7071). ∴ sin 9pi/4 = 0.7071.
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