Cos 360

The cosine of 360 degrees or cos 360 symbolizes the angle in the fourth quadrant. As we know, angle 360 is greater than 270 degrees and less than or equal to 360°. Also, 360 degrees indicates complete rotation in a xy plane. The value of the cos i.e. 270° to 360°, is always positive as the lie in the fourth quadrant and the value lies in the fourth quadrant is always positive. Therefore, cos 360 degree is also a positive value. The accurate value of cos 360 degrees is 1. Let us also learn the value of cos 180 here.


If we wish to determine cosine 360° value in radians, then the first step we need to perform is to multiply 360° by π/180.


Therefore, cos 360 = cos (360 * π/180) = cos 2π

Hence, we can write the value of cos 2π = 1


Here, π is represented for 180°, which is half of the rotation of a unit circle. Hence, 2π represents complete rotation for 360°. So, for any number of a complete  rotation say n, the value of cos will always remain equal to 1. Hence, cos 2nπ = 1.


Furthermore, we know that Cos(-(-θ)) = cos(θ), therefore, even if we move in the opposite direction, the value of cos 2nπ will always be equal.


What is the Value of Cos 360 Degrees?

The value of cos 360 is 1.


How to Determine Cos 360 Degrees Value?

As, we know the value of cos 360° is equal to 1. Now, let us know how we can calculate cos 360 value.


As we know, cos 0 degree equals 1.


Now, if we take off one complete rotation in a unit circle, we will come back to the initial point.


After the completion of one single rotation, the value of the angle will be determined as 360° or 2π in radians.


Hence, after getting back to the same position we will find

Cos 0° = cos 360°

Or

Cos 0° = 2π

Therefore, we can assume that,

Cos 360° = cos 2π = 1.


Important Identities of Cos 360 Degrees

  •  Cos 360 degree = sin (90° + 360°) = sin 450°

  •  cos 360 degree= sin (90°- 360°) = - sin 270°

  • - cos360 degree= cos (180°+ 360°) = cos 540°

  • - cos 360° degree = cos (180°- 360°) = - cos 180°

Let us look at the trigonometry ratio table which represent values both in radians and degree.


Trigonometry Ratio Table

Angles in Degrees

30°

45°

60°

90°

120°

150°

180°

210°

270°

300°

330°

360°

Angles in Radians

0

\[\frac{\pi}{6}\]

\[\frac{\pi}{4}\]

\[\frac{\pi}{3}\]

\[\frac{\pi}{2}\]

\[\frac{2\pi}{3}\]

\[\frac{5\pi}{6}\]

\[\pi\]

\[\frac{7\pi}{6}\]

\[\frac{3\pi}{2}\]

\[\frac{5\pi}{\sqrt{3}}\]

\[\frac{11\pi}{6}\]

\[2\pi\]

Sin

0

\[\frac{1}{2}\]

\[\frac{1}{\sqrt{2}}\]

\[\frac{\sqrt{3}}{2}\]

1

\[\frac{\sqrt{3}}{2}\]

\[\frac{1}{2}\]

0

\[\frac{-1}{2}\]

-1

\[-\frac{\sqrt{3}}{2}\]

\[\frac{-1}{2}\]

0

Cos

1

\[\frac{\sqrt{3}}{2}\]

\[\frac{1}{\sqrt{2}}\]

\[\frac{1}{2}\]

0

\[\frac{-1}{2}\]

\[-\frac{\sqrt{3}}{2}\]

-1

\[-\frac{\sqrt{3}}{2}\]

0

\[\frac{1}{2}\]

\[\frac{\sqrt{3}}{2}\]

1

Tan

0

\[\frac{1}{\sqrt{3}}\]

1

\[\sqrt{3}\]

\[\sqrt{3}\]

\[\frac{-1}{\sqrt{3}}\]

0

\[\frac{1}{\sqrt{3}}\]

\[-\sqrt{3}\]

\[\frac{-1}{\sqrt{3}}\]

0

Cot

\[\sqrt{3}\]

1

\[\frac{1}{\sqrt{3}}\]

0

\[\frac{-1}{\sqrt{3}}\]

\[-\sqrt{3}\]

\[-\sqrt{3}\]

0

\[-\sqrt{3}\]

Cosec

2

\[\sqrt{2}\]

\[\frac{2}{\sqrt{3}}\]

1

\[\frac{2}{\sqrt{3}}\]

2

-2

-1

\[\frac{-2}{\sqrt{3}}\]

-2

Sec

1

\[\frac{2}{\sqrt{3}}\]

\[\sqrt{2}\]

2

-2

\[\frac{-2}{\sqrt{3}}\]

-1

\[\frac{-2}{\sqrt{3}}\]

2

\[\frac{-2}{\sqrt{3}}\]

1


Solved Examples

1. Verify that cos (360°- y) = cos y

Solution: 

cos (360°- y) = cos 360° cos y + sin 360° sin y

cos (360°- y) = (1)cos y + (0) sin y

cos (360°- y) = cos y.


2. Verify that cos (180°+ x) = -cos x

Solution: 

cos (180°+ x) = cos 180° cos x + sin 180° sin x

cos (180°+ x) = (1)cos x - (0) sin x

cos (180°+ x) = -cos x.


3. Determine the following values 

  1. Sin 120°, 

  1. cos 120° 

  2. tan 120°

Solution: 

Sin 120° = Sin(180°- 120°) = \[\frac{\sqrt{3}}{2}\]

Cos 120° = - cos(180° - 120°) = \[-\frac{1}{2}\]

Tan 120° = Sin 120°/ Cos 120°

\[(\frac{\sqrt{3}}{2})(\frac{-1}{2}) = -\sqrt{3}\].


Quiz Time

1. What is the value of cos 420°?

  1. \[\frac{1}{2}\]

  2. 1

  3. 0

  4. \[\frac{\sqrt{3}}{2}\]


2. What is the value of cos 135°?

  1. \[\frac{1}{2}\]

  2. 1

  3. 0

  4. \[\frac{-1}{\sqrt{2}}\]

FAQ (Frequently Asked Questions)

1. Explain the Cosine Function.

The cosine function is one of the most commonly used trigonometric functions other than the sine and tangent function In a right angle triangle, cosine function is defined as the ratio of the length of the adjacent side of the right-angle triangle to its hypotenuse side. 


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For example, a triangle ABC with an angle alpha, the cosine function will be expressed as :

Cosine α - AdjacentSide/ Hypotenuse side

For the above triangle, the cosine function will be defined as 

Cosine α - AC/AB

Hence, the cosine formula for the above triangle is 

Cosine α - b/h.

2. Explain the Law of Cosine and its Formulas.

The law of cosines or cos law is one of the important trigonometric laws which is used to determine the unknown angles or sides of a triangle. The values of the angles can easily be calculated with the help of the cosine law if all the three sides of the right-angle triangle(SSS) are given. Similarly, the law of cosine is also used to determine the sides of the triangle, when the values of angles are already mentioned in the question.


Law of Cosine Formula

We can find the length of the sides of a right-angle triangle by using the following cosine law formula.

m² = n² + o² - 2no cos (O)

n² = m² + o² - 2mo cos (M)

o² = m² + n² - 2mn cos (N)


Following cosine law formula will be used if we want to find the angles of a right-angle triangle,

Cos O = (n + o - m)/2no

Cos M = ( m + o - n)/ 2mo

Cos N  = ( m + n+ o) /2mn