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:
- ± √(1-cos²(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 x-axis.
- The sin of 5pi/4 equals the y-coordinate(-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:
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:
- ± √(1-cos²(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 x-axis, 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 y-coordinate (-0.7071). ∴ sin 5pi/4 = -0.7071.