Cos 4pi

The value of cos 4pi is 1. Cos 4pi radians in degrees is written as cos ((4π) × 180°/π), i.e., cos (720°). In this article, we will discuss the methods to find the value of cos 4pi with examples.

• Cos 4pi: 1
• Cos (-4pi): 1
• Cos 4pi in degrees: cos (720°)

What is the Value of Cos 4pi?

The value of cos 4pi is 1. Cos 4pi can also be expressed using the equivalent of the given angle (4pi) in degrees (720°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 4pi radians = 4pi × (180°/pi) = 720° or 720 degrees
∴ cos 4pi = cos 4π = cos(720°) = 1

Explanation:

For cos 4pi, the angle 4pi > 2pi. We can represent cos 4pi as, cos(4pi mod 2pi) = cos(0). For cos 4pi, the angle 4pi lies on the positive x-axis. Thus, cos 4pi value = 1
Since the cosine function is a periodic function, we can represent cos 4pi as, cos 4pi = cos(4pi + n × 2pi), n ∈ Z.
⇒ cos 4pi = cos 6pi = cos 8pi , and so on.
Note: Since, cosine is an even function, the value of cos(-4pi) = cos(4pi).

Methods to Find Value of Cos 4pi

The value of cos 4pi is given as 1. We can find the value of cos 4pi by: 

• Using Unit Circle
• Using Trigonometric Functions

Cos 4pi Using Unit Circle

To find the value of cos 4π using the unit circle, represent 4pi in the form (2 × 2pi) + 0 [∵ 4pi>2pi] ∵ cosine is a periodic function, cos 4pi = cos 0.

• Rotate ‘r’ anticlockwise to form 0 or 4pi angle with the positive x-axis.
• The cos of 4pi equals the x-coordinate(1) of the point of intersection (1, 0) of unit circle and r.

Hence the value of cos 4pi = x = 1

Cos 4pi in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the cos 4pi as:

• ± √(1-sin²(4pi))
• ± 1/√(1 + tan²(4pi))
• ± cot(4pi)/√(1 + cot²(4pi))
• ±√(cosec²(4pi) - 1)/cosec(4pi)
• 1/sec(4pi)

Note: Since 4pi lies on the positive x-axis, the final value of cos 4pi is 1.

We can use trigonometric identities to represent cos 4pi as,

• -cos(pi - 4pi) = -cos(-3pi)
• -cos(pi + 4pi) = -cos 5pi
• sin(pi/2 + 4pi) = sin 9pi/2
• sin(pi/2 - 4pi) = sin(-7pi/2)

☛ Also Check:

• sin 7pi/12
• tan 3pi/4
• sin 5pi
• sin 11pi/12
• sec 3pi/4
• sin 4pi/3

FAQs on Cos 4pi

What is Cos 4pi?

Cos 4pi is the value of cosine trigonometric function for an angle equal to 4π radians. The value of cos 4pi is 1.

How to Find the Value of Cos 4pi?

The value of cos 4pi can be calculated by constructing an angle of 4π radians with the x-axis, and then finding the coordinates of the corresponding point (1, 0) on the unit circle. The value of cos 4pi is equal to the x-coordinate (1). ∴ cos 4pi = 1.

What is the Value of Cos 4pi in Terms of Sec 4pi?

Since the secant function is the reciprocal of the cosine function, we can write cos 4pi as 1/sec(4pi). The value of sec 4pi is equal to 1.

How to Find Cos 4pi in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of cos 4pi can be given in terms of other trigonometric functions as:

• ± √(1-sin²(4pi))
• ± 1/√(1 + tan²(4pi))
• ± cot(4pi)/√(1 + cot²(4pi))
• ±√(cosec²(4pi) - 1)/cosec(4pi)
• 1/sec(4pi)

☛ Also check: trigonometric table

What is the Value of Cos 4pi in Terms of Cot 4pi?

We can represent the cosine function in terms of the cotangent function using trig identities, cos 4pi can be written as cot(4pi)/√(1 + cot²(4pi)).