Cos 2pi/8
The value of cos 2pi/8 is 0.7071067. . .. Cos 2pi/8 radians in degrees is written as cos ((2π/8) × 180°/π), i.e., cos (45°). In this article, we will discuss the methods to find the value of cos 2pi/8 with examples.
 Cos 2pi/8: 1/√2
 Cos 2pi/8 in decimal: 0.7071067. . .
 Cos (2pi/8): 0.7071067. . . or 1/√2
 Cos 2pi/8 in degrees: cos (45°)
What is the Value of Cos 2pi/8?
The value of cos 2pi/8 in decimal is 0.707106781. . .. Cos 2pi/8 can also be expressed using the equivalent of the given angle (2pi/8) in degrees (45°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 2pi/8 radians = 2pi/8 × (180°/pi) = 45° or 45 degrees
∴ cos 2pi/8 = cos 2π/8 = cos(45°) = 1/√2 or 0.7071067. . .
Explanation:
For cos 2pi/8, the angle 2pi/8 lies between 0 and pi/2 (First Quadrant). Since cosine function is positive in the first quadrant, thus cos 2pi/8 value = 1/√2 or 0.7071067. . .
Since the cosine function is a periodic function, we can represent cos 2pi/8 as, cos 2pi/8 = cos(2pi/8 + n × 2pi), n ∈ Z.
⇒ cos 2pi/8 = cos 9pi/4 = cos 17pi/4 , and so on.
Note: Since, cosine is an even function, the value of cos(2pi/8) = cos(2pi/8).
Methods to Find Value of Cos 2pi/8
The cosine function is positive in the 1st quadrant. The value of cos 2pi/8 is given as 0.70710. . .. We can find the value of cos 2pi/8 by:
 Using Trigonometric Functions
 Using Unit Circle
Cos 2pi/8 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 2pi/8 as:
 ± √(1sin²(2pi/8))
 ± 1/√(1 + tan²(2pi/8))
 ± cot(2pi/8)/√(1 + cot²(2pi/8))
 ±√(cosec²(2pi/8)  1)/cosec(2pi/8)
 1/sec(2pi/8)
Note: Since 2pi/8 lies in the 1st Quadrant, the final value of cos 2pi/8 will be positive.
We can use trigonometric identities to represent cos 2pi/8 as,
 cos(pi  2pi/8) = cos 3pi/4
 cos(pi + 2pi/8) = cos 5pi/4
 sin(pi/2 + 2pi/8) = sin 3pi/4
 sin(pi/2  2pi/8) = sin pi/4
Cos 2pi/8 Using Unit Circle
To find the value of cos 2π/8 using the unit circle:
 Rotate ‘r’ anticlockwise to form 2pi/8 angle with the positive xaxis.
 The cos of 2pi/8 equals the xcoordinate(0.7071) of the point of intersection (0.7071, 0.7071) of unit circle and r.
Hence the value of cos 2pi/8 = x = 0.7071 (approx)
☛ Also Check:
Examples Using Cos 2pi/8

Example 1: Find the value of 2 cos(2pi/8)/3 sin(pi/4).
Solution:
Using trigonometric identities, we know, cos(2pi/8) = sin(pi/2  2pi/8) = sin pi/4.
⇒ cos(2pi/8) = sin(pi/4)
⇒ Value of 2 cos(2pi/8)/3 sin(pi/4) = 2/3 
Example 2: Using the value of cos 2pi/8, solve: (1sin²(2pi/8)).
Solution:
We know, (1sin²(2pi/8)) = (cos²(2pi/8)) = 0.5
⇒ (1sin²(2pi/8)) = 0.5 
Example 3: Find the value of (cos² pi/8  sin² pi/8). [Hint: Use cos 2pi/8 = 0.7071]
Solution:
Using the cos 2a formula,
(cos² pi/8  sin² pi/8) = cos(2 × pi/8) = cos 2pi/8
∵ cos 2pi/8 = 0.7071
⇒ (cos² pi/8  sin² pi/8) = 0.7071
FAQs on Cos 2pi/8
What is Cos 2pi/8?
Cos 2pi/8 is the value of cosine trigonometric function for an angle equal to 2π/8 radians. The value of cos 2pi/8 is 1/√2 or 0.7071 (approx)
What is the Exact Value of cos 2pi/8?
The exact value of cos 2pi/8 can be given accurately up to 8 decimal places as 0.70710678 and 1/√2 in fraction.
How to Find the Value of Cos 2pi/8?
The value of cos 2pi/8 can be calculated by constructing an angle of 2π/8 radians with the xaxis, and then finding the coordinates of the corresponding point (0.7071, 0.7071) on the unit circle. The value of cos 2pi/8 is equal to the xcoordinate (0.7071). ∴ cos 2pi/8 = 0.7071.
How to Find Cos 2pi/8 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 2pi/8 can be given in terms of other trigonometric functions as:
 ± √(1sin²(2pi/8))
 ± 1/√(1 + tan²(2pi/8))
 ± cot(2pi/8)/√(1 + cot²(2pi/8))
 ±√(cosec²(2pi/8)  1)/cosec(2pi/8)
 1/sec(2pi/8)
☛ Also check: trigonometric table
What is the Value of Cos 2pi/8 in Terms of Tan 2pi/8?
We know, using trig identities, we can write cos 2pi/8 as 1/√(1 + tan²(2pi/8)). Here, the value of tan 2pi/8 is equal to 1.
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