An Introduction
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.
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°
\[\pm \sqrt{(1-sin^{2}(360^{\circ}))}\]
\[\frac{1}{\pm \sqrt{(1+tan^{2}(360^{\circ}))}}\]
\[\frac{cot 360^{\circ}}{\pm \sqrt{(1+cot^{2}(360^{\circ}))}}\]
\[\pm \frac{\sqrt{(cosec^{2}(360^{\circ}))-1}}{cosec 360}\]
\[\frac{1}{sec 360^{\circ}}\]
Let us look at the trigonometry ratio table which represents values both in radians and degrees.
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
Sin 120°,
cos 120°
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°?
\[\frac{1}{2}\]
1
0
\[\frac{\sqrt{3}}{2}\]
2. What is the value of cos 135°?
\[\frac{1}{2}\]
1
0
\[\frac{-1}{\sqrt{2}}\]
FAQs on Cos 360°
1. Explain the Cosine Function.
To find the value of 360°, the first step is to construct a 360° angle with the X-axis. Then, the coordinates of the corresponding points are determined which are (1,0) on the unit circle. It is observed that the value of the angle of 360° is equal to the coordinate on the x-axis which is equal to one. Therefore, we conclude that the value of the cosine of 360° is equal to one.
It is also written as cos 360°=1.
2. How are the mathematical values of different trigonometric functions calculated?
To find the mathematical values of different trigonometric, let us have a look at the formulas given in the below table-
S.no | Property | Value | Relation with other Trigonometric Functions |
1 | sin A | \[\frac{perpendicular}{Hypotenues}\] | |
2 | cos A | \[\frac{Base}{Hypotenues}\] | |
3 | tan A | \[\frac{perpendicular}{Base}\] | \[\frac{SinA}{Cos A}\] |
4 | cot A | \[\frac{Base}{perpendicular}\] | \[\frac{Cos A}{Sin A}\] |
5 | cosec A | \[\frac{hypotenues}{perpendicular}\] | \[\frac{1}{Sin A}\] |
6 | sec A | \[\frac{hypotenues}{base}\] | \[\frac{1}{Cos A}\] |
3. What are the real-life examples of trigonometry?
Trigonometry is applied to numerous aspects in life such as-
If the sun’s rays are cast upon the stick of a tree and its shadow at the same time, we can calculate the length of the tree as well as of the stick.
If a tank and a fighter plane are nearby, then the soldier driving the tank can determine his distance from the plane and the height of the plane above the ground at the same time. With the help of this information, he can figure out the angle between him and the plane and can shoot the plane.
4. What are the applications of trigonometry in real life?
Trigonometry has a wide number of applications in real life such as-
While making a roof to make the roof inclined and to determine the height of the roof in a building, trigonometry can be used.
Trigonometry is also used by the aviation and the naval industry.
It is used by cartographers to create maps.
It is used to determine the distance between near and far objects.
It is used in archaeology, flight engineering and oceanography.
5. What are the Pythagorean identities for trigonometry?
Since in a right-angled triangle, the legs are known by the values of sin and cos , using the Pythagorean theorem we can obtain
sin2+cos2=1
Equation 1
Using this identity we determine two more identities.
sin2+cos2=1
Diving each term by cos2
sin2cos2+cos2cos2=1cos2
We get
tan2+1=sec2
Equation 2
Now by dividing eq. 1 by sin2
sin2sin2+cos2sin2=1sin2
We get
1+cot2=cosec2
Equation 3
To learn more about trigonometric identities, students can visit Vedantu’s study material on Trigonometry.
6. 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, the cosine function is defined as the ratio of the length of the adjacent side of the right-angle triangle to its hypotenuse side.
(the image will be uploaded soon)
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.
7. 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
