Cos 360°

The functions in trigonometry that relate an angle of a right-angled triangle to ratios of two side lengths are known as trigonometric functions. There are 6 trigonometric ratios which are sine, cosecant, tangent, cosine, secant and cotangent.

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 they 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 the 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.

• In trigonometry, the value of cosine 360° is 1. We know that 

\[\theta in radians = \theta degree \times \frac{\pi }{180^{\circ}}\]

• Therefore, cos (360°) can also be written in radians as\[cos(360^{\circ} \times \frac{\pi }{180^{\circ}})\] which means cos(2) or cos(6.2831...). 

• The value of cos(-360°) is also 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 the 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.

Trigonometry Ratio Table

Solved Examples related to cos 360°

1. Determine the value of (cos2180°-sin2180°).

Ans: Using cos 2a formula,

\[(cos^{2}180^{\circ}-sin^{2}180^{\circ})=cos(2\times 180^{\circ})=cos 360^{\circ}\]

We know that cos 360°=1

\[(cos^{2}180^{\circ}-sin^{2}180^{\circ})= 1\]

2. Determine the value of \[\frac{2cos(360^{\circ})}{3sin(-270^{\circ})}\].

Ans: We know, 

\[cos 360^{\circ}=sin(90^{\circ}-360^{\circ})\]

\[cos 360^{\circ}=sin(-270^{\circ})\]

Value of  \[\frac{2cos(360^{\circ})}{3sin(-270^{\circ})}\]= \[\frac{2}{3}\]

3. Verify that cos (360°- y) = cos y

Ans: 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.

4. Verify that cos (180°+ x) = -cos x

Ans: 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.

5. Determine the following values 

1. Sin 120°, 

1. cos 120° 

2. tan 120°

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

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

Tan 120° = Sin 120°/ Cos 120°=\[\frac{\sqrt{3}}{2}\]\[\frac{1}{2}\]=-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. \[\frac{-1}{\sqrt{2}}\]