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:

• ± √(1-sin²(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 x-axis.
• The cos of 2pi/8 equals the x-coordinate(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:

• sin 5pi/4
• cos pi/3
• cot 2pi
• sin 2pi/3
• cot pi/6
• cos 7pi/4

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 x-axis, 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 x-coordinate (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:

• ± √(1-sin²(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.