Cos 2pi/7
The value of cos 2pi/7 is 0.6234898. . .. Cos 2pi/7 radians in degrees is written as cos ((2π/7) × 180°/π), i.e., cos (51.4285714°). In this article, we will discuss the methods to find the value of cos 2pi/7 with examples.
- Cos 2pi/7 in decimal: 0.6234898. . .
- Cos (-2pi/7): 0.6234898. . .
- Cos 2pi/7 in degrees: cos (51.4285714°)
What is the Value of Cos 2pi/7?
The value of cos 2pi/7 in decimal is 0.623489802. . .. Cos 2pi/7 can also be expressed using the equivalent of the given angle (2pi/7) in degrees (51.4285714°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 2pi/7 radians = 2pi/7 × (180°/pi) = 51.4285714° or 51.4285714 degrees
∴ cos 2pi/7 = cos 2π/7 = cos(51.4285714°) = 0.6234898. . .
Explanation:
For cos 2pi/7, the angle 2pi/7 lies between 0 and pi/2 (First Quadrant). Since cosine function is positive in the first quadrant, thus cos 2pi/7 value = 0.6234898. . .
Since the cosine function is a periodic function, we can represent cos 2pi/7 as, cos 2pi/7 = cos(2pi/7 + n × 2pi), n ∈ Z.
⇒ cos 2pi/7 = cos 16pi/7 = cos 30pi/7 , and so on.
Note: Since, cosine is an even function, the value of cos(-2pi/7) = cos(2pi/7).
Methods to Find Value of Cos 2pi/7
The cosine function is positive in the 1st quadrant. The value of cos 2pi/7 is given as 0.62348. . .. We can find the value of cos 2pi/7 by:
- Using Trigonometric Functions
- Using Unit Circle
Cos 2pi/7 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 2pi/7 as:
- ± √(1-sin²(2pi/7))
- ± 1/√(1 + tan²(2pi/7))
- ± cot(2pi/7)/√(1 + cot²(2pi/7))
- ±√(cosec²(2pi/7) - 1)/cosec(2pi/7)
- 1/sec(2pi/7)
Note: Since 2pi/7 lies in the 1st Quadrant, the final value of cos 2pi/7 will be positive.
We can use trigonometric identities to represent cos 2pi/7 as,
- -cos(pi - 2pi/7) = -cos 5pi/7
- -cos(pi + 2pi/7) = -cos 9pi/7
- sin(pi/2 + 2pi/7) = sin 11pi/14
- sin(pi/2 - 2pi/7) = sin 3pi/14
Cos 2pi/7 Using Unit Circle
To find the value of cos 2π/7 using the unit circle:
- Rotate ‘r’ anticlockwise to form 2pi/7 angle with the positive x-axis.
- The cos of 2pi/7 equals the x-coordinate(0.6235) of the point of intersection (0.6235, 0.7818) of unit circle and r.
Hence the value of cos 2pi/7 = x = 0.6235 (approx)
☛ Also Check:
FAQs on Cos 2pi/7
What is Cos 2pi/7?
Cos 2pi/7 is the value of cosine trigonometric function for an angle equal to 2π/7 radians. The value of cos 2pi/7 is 0.6235 (approx).
What is the Value of Cos 2pi/7 in Terms of Cosec 2pi/7?
Since the cosine function can be represented using the cosecant function, we can write cos 2pi/7 as [√(cosec²(2pi/7) - 1)/cosec 2pi/7]. The value of cosec 2pi/7 is equal to 1.27904.
How to Find Cos 2pi/7 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 2pi/7 can be given in terms of other trigonometric functions as:
- ± √(1-sin²(2pi/7))
- ± 1/√(1 + tan²(2pi/7))
- ± cot(2pi/7)/√(1 + cot²(2pi/7))
- ±√(cosec²(2pi/7) - 1)/cosec(2pi/7)
- 1/sec(2pi/7)
☛ Also check: trigonometric table
How to Find the Value of Cos 2pi/7?
The value of cos 2pi/7 can be calculated by constructing an angle of 2π/7 radians with the x-axis, and then finding the coordinates of the corresponding point (0.6235, 0.7818) on the unit circle. The value of cos 2pi/7 is equal to the x-coordinate (0.6235). ∴ cos 2pi/7 = 0.6235.
What is the Value of Cos 2pi/7 in Terms of Sin 2pi/7?
Using trigonometric identities, we can write cos 2pi/7 in terms of sin 2pi/7 as, cos(2pi/7) = √(1 - sin²(2pi/7)). Here, the value of sin 2pi/7 is equal to 0.7818.