Sin 2pi/5
The value of sin 2pi/5 is 0.9510565. . .. Sin 2pi/5 radians in degrees is written as sin ((2π/5) × 180°/π), i.e., sin (72°). In this article, we will discuss the methods to find the value of sin 2pi/5 with examples.
- Sin 2pi/5: √(10 - 2√5)/2
- Sin 2pi/5 in decimal: 0.9510565. . .
- Sin (-2pi/5): -0.9510565. . .
- Sin 2pi/5 in degrees: sin (72°)
What is the Value of Sin 2pi/5?
The value of sin 2pi/5 in decimal is 0.951056516. . .. Sin 2pi/5 can also be expressed using the equivalent of the given angle (2pi/5) in degrees (72°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 2pi/5 radians = 2pi/5 × (180°/pi) = 72° or 72 degrees
∴ sin 2pi/5 = sin 2π/5 = sin(72°) = √(10 - 2√5)/2 or 0.9510565. . .
Explanation:
For sin 2pi/5, the angle 2pi/5 lies between 0 and pi/2 (First Quadrant). Since sine function is positive in the first quadrant, thus sin 2pi/5 value = √(10 - 2√5)/2 or 0.9510565. . .
Since the sine function is a periodic function, we can represent sin 2pi/5 as, sin 2pi/5 = sin(2pi/5 + n × 2pi), n ∈ Z.
⇒ sin 2pi/5 = sin 12pi/5 = sin 22pi/5 , and so on.
Note: Since, sine is an odd function, the value of sin(-2pi/5) = -sin(2pi/5).
Methods to Find Value of Sin 2pi/5
The sine function is positive in the 1st quadrant. The value of sin 2pi/5 is given as 0.95105. . .. We can find the value of sin 2pi/5 by:
- Using Trigonometric Functions
- Using Unit Circle
Sin 2pi/5 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 2pi/5 as:
- ± √(1-cos²(2pi/5))
- ± tan(2pi/5)/√(1 + tan²(2pi/5))
- ± 1/√(1 + cot²(2pi/5))
- ± √(sec²(2pi/5) - 1)/sec(2pi/5)
- 1/cosec(2pi/5)
Note: Since 2pi/5 lies in the 1st Quadrant, the final value of sin 2pi/5 will be positive.
We can use trigonometric identities to represent sin 2pi/5 as,
- sin(pi - 2pi/5) = sin 3pi/5
- -sin(pi + 2pi/5) = -sin 7pi/5
- cos(pi/2 - 2pi/5) = cos pi/10
- -cos(pi/2 + 2pi/5) = -cos 9pi/10
Sin 2pi/5 Using Unit Circle
To find the value of sin 2π/5 using the unit circle:
- Rotate ‘r’ anticlockwise to form 2pi/5 angle with the positive x-axis.
- The sin of 2pi/5 equals the y-coordinate(0.9511) of the point of intersection (0.309, 0.9511) of unit circle and r.
Hence the value of sin 2pi/5 = y = 0.9511 (approx)
☛ Also Check:
FAQs on Sin 2pi/5
What is Sin 2pi/5?
Sin 2pi/5 is the value of sine trigonometric function for an angle equal to 2pi/5 radians. The value of sin 2pi/5 is √(10 - 2√5)/2 or 0.9511 (approx).
What is the Value of Sin 2pi/5 in Terms of Cosec 2pi/5?
Since the cosecant function is the reciprocal of the sine function, we can write sin 2pi/5 as 1/cosec(2pi/5). The value of cosec 2pi/5 is equal to 1.05146.
What is the Value of Sin 2pi/5 in Terms of Tan 2pi/5?
We know, using trig identities, we can write sin 2pi/5 as tan(2pi/5)/√(1 + tan²(2pi/5)). Here, the value of tan 2pi/5 is equal to 3.077683.
How to Find Sin 2pi/5 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 2π/5 can be given in terms of other trigonometric functions as:
- ± √(1-cos²(2pi/5))
- ± tan(2pi/5)/√(1 + tan²(2pi/5))
- ± 1/√(1 + cot²(2pi/5))
- ± √(sec²(2pi/5) - 1)/sec(2pi/5)
- 1/cosec(2pi/5)
☛ Also check: trigonometry table
How to Find the Value of Sin 2pi/5?
The value of sin 2pi/5 can be calculated by constructing an angle of 2π/5 radians with the x-axis, and then finding the coordinates of the corresponding point (0.309, 0.9511) on the unit circle. The value of sin 2pi/5 is equal to the y-coordinate (0.9511). ∴ sin 2pi/5 = 0.9511.