Cos 2pi
Cos 2pi gives the value of the cosine function when the angle made with the positive xaxis is 2pi, that is, one complete rotation. We study values of trigonometric functions at some standard angles like 0, π/6, π/4, π/3/ π/2 (in radians) and use them to solve various problems to determine the trigonometric values with nonstandard angles. The value of cos 2pi is one such value and can be found using different methods. In this article, we will determine the value of cos 2pi as 1 using various methods along with some examples.
1.  What is Cos 2pi? 
2.  Cos 2pi Using cos x Graph 
3.  Cos 2pi Using Unit Circle 
4.  Cos 2pi Using Standard Angles 
5.  Cos 2pi Using Double Angle Formula 
6.  FAQs on cos 2pi 
What is Cos 2pi?
Cos 2pi is the value of the cosine function cos x when x = 2π. We know that the period of the cosine function is 2π, that is, the cycle of cos x repeats after every 2π radians. Hence, if we subtract 2π from 2π, the value of cos x will be the same as cos 2pi and we get back to 0 radians. So, the value of cos 2pi is equal to the value of cos 0 which is 1. So, cos 2pi is equal to 1.
The trigonometric table gives the values of the trigonometric functions at standard (specific) angles such as 0, π/6, π/4, π/3/ π/2. Let us recall the values of the cosine function at these angles: cos 0 = 1, cos π/6 = √3/2, cos π/4 = √2/2, cos π/3 = 1/2, and cos π/2 = 0. Now, we will use some of these values to determine the value of cos 2pi using different methods such as graphically, standard angles, unit circle, and double angle formula.
Cos 2pi Using cos x Graph
We will plot the graph of the cosine function and check the value of cos x when x = 2π to determine the value of cos 2pi. Given below is the graph of cos x. We know that the period of cos x is 2pi and the graph repeats after every 2pi radians. Hence, the value of cos 2pi is the same as the value of cos 0 in the graph. Hence, the value of cos 2pi from the graph of cos x is 1.
Cos 2pi Using Unit Circle
We will draw a unit circle with a center at the origin and radius equal to 1. Now, we know that every point on the circumference of the unit circle has the coordinates (cosθ, sinθ), where θ is the angle made by the line segment joining the origin and the point on the circle with the positive direction of the xaxis and the angle is measured in the anticlockwise direction.
To determine the value of cos 2pi, we need to understand that the line which makes an angle of 2pi, that is, 360° is the xaxis itself because 2pi implies one complete rotation and when the xaxis makes one complete rotation, it comes back to the point from where it started, that is, when the angle was 0 radians. Also, as we can see that the point on the circle which makes angle 2pi radians has coordinates (1, 0) as 1 is the radius of the circle and the xintercept is 0. Hence, we have
(cos 2pi, sin 2pi) = (1, 0) ⇒ cos 2pi = xcoordinate of (1, 0) = 1
Cos 2pi Using Standard Angles
To determine the value of cos 2pi using standard angles, we need to reduce cos 2pi into cosine of one of the standard angles 0, π/6, π/4, π/3/ π/2. We know that the cosine function repeats after every 2π radians which makes it one complete rotation of the unit circle, that is, cos(2pi  x) = cos x. This implies we will subtract 2pi from 2pi, which gives us 2pi  2pi = 0. Hence, it either lies in the first quadrant or the fourth quadrant. Since, cosine function is positive in both first and fourth quadrants, we have cos 2pi = + cos 0 = 1
Cos 2pi Using Double Angle Formula
We can find the value of cos 2pi using the double angle formula of the cosine function, that is, cos 2x = cos^{2}x  sin^{2}x. We have cos 2pi = cos^{2}π  sin^{2}π. Since π is not a standard angle we will determine the values of cos π and sin π using angle sum formulas.
cos π = cos (π/2 + π/2) = cos π/2 cos π/2  sin π/2 sin π/2 = 0×0  1×1 = 1
sin π = sin (π/2 + π/2) = sin π/2 cos π/2 + sin π/2 sin π/2 = 1×0 + 1×0 = 0
Substitute these values in cos 2pi = cos^{2}π  sin^{2}π, we have cos 2pi = (1)^{2}  0 = 1. Hence cos 2 pi is equal to 1.
Important Notes on cos 2pi
 cos nπ = (1)^{n}, n is an integer
 The value of cos 2pi from the graph of cos x is 1.
 sin nπ = 0, n is an integer
Related Topics on cos 2pi
Examples on Cos 2pi

Example 1: Evaluate cos 2π cos π/6  cos 2π cos π/3 using the value of cos 2pi.
Solution: We know that cos π/6 = √3/2, cos π/3 = 1/2, cos 2π = 1. Substitute these values in cos 2π cos π/6  cos 2π cos π/3, we have
cos 2π cos π/6  cos 2π cos π/3 = 1 × √3/2  1 × 1/2 = √3/2  1/2 = (√3  1)/2
Answer: cos 2π cos π/6  cos 2π cos π/3 = (√3  1)/2

Example 2: Determine the value of sin 2pi using the value of cos 2pi.
Solution: We know that the value of cos 2pi is 1. We will use the trigonometric identity cos^{2}x + sin^{2}x = 1.
⇒ sin^{2}x = 1  cos^{2}x
⇒ sin^{2}2π = 1  cos^{2}2π = 1  1^{2 }= 1  1 = 0
Answer: Hence sin 2pi is equal to 0 using cos 2pi value.
FAQs on cos 2pi
What is the Value of Cos 2pi?
The value of cos 2pi is 1 which can be obtained using different methods.
How to Find the Value of cos 2pi?
The value of cos 2pi can be calculated using different methods such as graphically, standard angles, unit circle, and double angle formula.
How to Determine the Value of Tan 2pi Using Cos 2pi?
We know that sin 2pi is equal to 0 using the cos 2pi value and the value of cos 2pi is equal to 1. Also, tan x = sin x/cos x, therefore, tan 2pi = sin 2pi/ cos 2pi = 0/1 = 0. Hence tan 2pi is equal to 0.
Is the Value of cos pi equal to the Value of cos 2pi?
No, the value of cos pi is NOT equal to the value of cos 2pi as cos pi is equal to 1 and cos 2pi is equal to 1.
What is the Value of sin 2pi Using cos 2pi?
We know the trigonometric identity cos^{2}x + sin^{2}x = 1 ⇒ sin^{2}x = 1  cos^{2}x ⇒ sin^{2}2π = 1  cos^{2}2π = 1  1^{2 }= 1  1 = 0. Hence sin 2pi is equal to 0 using cos 2pi value.
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